Regularity of Einstein Manifolds
and the Codimension $4$ Conjecture

Jeff Cheeger and Aaron Naber

Date: August 24, 2026

Abstract.

In this paper, we are concerned with the regularity of noncollapsed Riemannian manifolds $(M^{n},g)$ with bounded Ricci curvature, as well as their Gromov-Hausdorff limit spaces $(M^{n}_{j},d_{j})\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}(X,d)$ , where $d_{j}$ denotes the Riemannian distance. Our main result is a solution to the codimension $4$ conjecture, namely that $X$ is smooth away from a closed subset of codimension $4$ . We combine this result with the ideas of quantitative stratification to prove a priori $L^{q}$ estimates on the full curvature $|{\rm Rm}|$ for all $q<2$ . In the case of Einstein manifolds, we improve this to estimates on the regularity scale. We apply this to prove a conjecture of Anderson that the collection of $4$ -manifolds $(M^{4},g)$ with $|{\rm Ric}_{M^{4}}|\leq 3$ , ${\rm Vol}(M)>{\rm v}>0$ , and ${\rm diam}(M)\leq D$ contains at most a finite number of diffeomorphism classes. A local version of this is used to show that noncollapsed $4$ -manifolds with bounded Ricci curvature have a priori $L^{2}$ Riemannian curvature estimates.

[01XD]

1. Introduction

In this paper, we consider pointed Riemannian manifolds $(M^{n},g,p)$ with bounded Ricci curvature

\displaystyle|{\rm Ric}_{M^{n}}|\leq n-1\,,

(1.1)

which satisfy the noncollapsing assumption

\displaystyle{\rm Vol}(B_{1}(p))>{\rm v}>0\,.

(1.2)

We will be particularly concerned with pointed Gromov-Hausdorff limits

\displaystyle(M^{n}_{j},d_{j},p_{j})\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}(X,d,p)

(1.3)

of sequences of such manifolds, where $d_{j}$ always denotes the Riemannian distance. Our main result is that $X$ is smooth away from a closed subset of codimension $4$ .¹¹ 1 In the Kähler case, this was shown in [Ch2], and independently by Tian, by exploiting the first Chern form and its relation to Ricci curvature. We will combine this with the previous work of the authors on quantitative stratification to show that $X$ satisfies a priori $L^{q}$ -estimates on the curvature $|{\rm Rm}|$ for all $q<2$ ; see Theorems 1.1 and 1.3. Finally, we will apply the results in the dimension $4$ setting in which there are various improvements, including a finiteness theorem up to diffeomorphism and an a priori $L^{2}$ curvature bound, for noncollapsed manifolds with bounded Ricci curvature; see Theorems 1.4 and 1.5.

The first major results on limit spaces satisfying (1.1)–(1.3) were proved in the $4$ -dimensional case. They made the additional assumptions that the $M^{4}_{j}$ have bounded diameter and Betti numbers [BKN89], [B90], [T90]. A key ingredient of the early results is that under these assumptions it follows directly from the Chern-Gauss-Bonnet formula for the Euler characteristic that the $L^{2}$ -norm of the curvature is bounded. By combining this with the appropriate $\epsilon$ -regularity results it is eventually proved that under the assumed topological constraints, any limit space must be an orbifold. This is carried further in [A90], where it is shown that the collection of noncollapsed Einstein $4$ -manifolds with bounded diameter and bounded Betti numbers have only finitely many diffeomorphism types. It was conjectured in [A94] that the Betti number bound was an unnecessary assumption. In Theorem 1.4. we prove this conjecture.

In higher dimensions, the study of Gromov-Hausdorff limit spaces satisfying (1.1), (1.2), was originally only possible under the additional assumption of $L^{q}$ -bounds on the curvature operator, see for instance [AnCh2], [CCT02]. The first step toward the study such limits without the need for curvature assumptions was taken in [ChCo1], where a stratification theory for noncollapsed limits with only lower Ricci curvature bounds was developed. By combining this with the $\epsilon$ -regularity results of [A90] one could then prove that noncollapsed limits of manifolds (1.3) with bounded Ricci curvature are smooth outside a closed subset of codimension $2$ . More recently, it was shown in [ChNa13] that one can then prove a priori $L^{q}$ -bounds on the curvature for all $q<1$ .

Based on knowledge of the $4$ -dimensional case, early workers conjectured that the singular set of noncollapsed limits of the form (1.3) should form a closed subset of codimension $4$ ; compare [A90], [B90], [T90]. The following is the main result of this paper:

[01XE]

Theorem 1.1.

Let $(M^{n}_{j},d_{j},p_{j})\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}(X,d,p)$ be a Gromov-Hausdorff limit of manifolds with $|{\rm Ric}_{M^{n}_{j}}|\leq n-1$ and ${\rm Vol}(B_{1}(p_{j}))>{\rm v}>0$ . Then the singular set $\mathcal{S}$ satisfies

\displaystyle\dim\mathcal{S}\leq n-4.

(1.4)

The dimension can be taken to be the Hausdorff or Minkowski dimension.

We will outline the proof of Theorem 1.1 in subsection 1.1. First we will discuss various applications. Our first applications are to the regularity theory of Einstein manifolds. To make this precise, let us begin with the following definition, see also [ChNa13]:

[01XF]

Definition 1.2.

For $x\in X$ we define the regularity scale $r_{x}$ by

\displaystyle r_{x}\equiv\max_{0<r\leq 1}\big\{\sup_{B_{r}(x)}|{\rm Rm}|\leq r^{-2}\big\}\,.

(1.5)

If $x\in\mathcal{S}$ is in the singular set of $X$ , then $r_{x}\equiv 0$ .

Let $T_{r}(S)=\{x\in M:d(x,S)<r\}$ denote the $r$ -tube around the set $S$ . By combining Theorem 1.1 with the quantitative stratification ideas of [ChNa13], we can show the following:

[01XG]

Theorem 1.3.

There exists $C=C(n,{\rm v},q)$ such that if $M^{n}$ satisfies $|{\rm Ric}_{M^{n}}|\leq n-1$ and ${\rm Vol}(B_{1}(p))>{\rm v}>0$ . then for each $q<2$ ,

\displaystyle\fint_{B_{1}(p)}|{\rm Rm}|^{q}\leq C\,.

(1.6)

If in addition, $M^{n}$ is assumed to be Einstein, then for every $q<2$ we have that

\displaystyle{\rm Vol}(T_{r}(\{x\in B_{1}(p):r_{x}\leq r\}))\leq C\,r^{2q}

(1.7)

[01XH]

Remark 1.1.

We can replace the assumption that $M^{n}$ is Einstein with just a bound on $|\nabla{\rm Ric}_{M^{n}}|$ to obtain the same result. In fact, if we only assume a bound on the Ricci curvature $|{\rm Ric}_{M^{n}}|$ , then (1.7) holds with the regularity scale $r_{x}$ replaced by the harmonic radius $r_{h}$ , see Definition 2.2. Note that estimates on the regularity scale are much stronger than corresponding $L^{q}$ estimates for the curvature given in (1.6).

The final theorems of the paper concern the $4$ -dimensional case in which we can make some marked improvements on the results in the general case. Let us begin with the following, which is a conjecture of Anderson [A94].

[01XI]

Theorem 1.4.

There exists $C=C({\rm v},D)$ such that if $M^{4}$ satisfies $|{\rm Ric}_{M^{4}}|\leq 3$ , ${\rm Vol}(B_{1}(p))>{\rm v}>0$ and ${\rm diam}(M^{n})\leq D$ , then $M^{4}$ can have one of at most $C$ diffeomorphism types.

By proving a more local version of the above theorem, we can improve Theorem 1.3 in the $4$ -dimensional case and show that the $L^{q}$ bounds on the curvature for $q<2$ may be pushed all the way to an a priori $L^{2}$ bound in dimension $4$ . We conjecture in Section 9 that this holds in all dimensions.

[01XJ]

Theorem 1.5.

There exists $C=C({\rm v})$ such that if $M^{4}$ satisfies $|{\rm Ric}_{M^{4}}|\leq 3$ and ${\rm Vol}(B_{1}(p))>{\rm v}>0$ , then

\displaystyle\fint_{B_{1}(p)}|{\rm Rm}|^{2}\leq C\,.

(1.8)

Furthermore, we have the sharp weak- $L^{2}$ estimate on the harmonic radius,

\displaystyle{\rm Vol}(T_{r}(\{x\in B_{1}(p):r_{h}\leq r\}))\leq Cr^{4}\,.

(1.9)

If we assume in addition that $M^{4}$ is Einstein, then the same result holds with the harmonic radius $r_{h}$ replaced by the regularity scale $r_{x}$ .

[01XK]

Remark 1.2.

If the assumption that $M^{4}$ is Einstein is weakened to assuming a bound on $|\nabla{\rm Ric}_{M^{n}}|$ , then (1.9) still holds with the harmonic radius $r_{h}$ replaced by the stronger regularity scale $r_{x}$ .

Next, we will give a brief outline of the paper. We begin in subsection 1.1 by outlining the proof of Theorem 1.1. This includes statements and explanations of some of the main technical theorems of the paper.

In Section 2 we go over some basic background and preliminary material. This includes the basics of stratifications for limit spaces, the standard $\epsilon$ -regularity theorem for spaces with bounded Ricci curvature, and some motivating examples.

Sections 3 and 4 are the the most crucial sections of the paper. There, we prove Theorems 1.11 and Theorem 1.8, the Transformation and Slicing theorems which, roughly speaking, allow us to blow up along a collection of points which is large enough to see into the singular set; see Section 1.1 for more on this.

Section 5 is dedicated to proving the main result of the paper, Theorem 1.1. The argument is a blow up argument that exploits the Slicing Theorem of Section 4. In Section 6, based on Theorem 1.1, we give a new $\epsilon$ -regularity theorem. Theorem 6.1 states that if a ball in a space with bounded Ricci curvature is close enough in the Gromov-Hausdorff sense to a ball in a metric cone, $\text{R}^{n-3}\times C(Z)$ , then the concentric ball of half the radius must be smooth.

In Section 7, the $\epsilon$ -regularity theorem of Section 6 is combined with the ideas of quantitative stratification to give effective improvements on all the results of the paper. We show that the singular set has codimension $4$ in the Minkowski sense, and give effective estimates for tubes around the balls of curvature concentration. This culminates in the proof of Theorem 1.3. In subection 7.2, we use the effective estimates of Theorem 1.3 to prove new estimates for harmonic functions on spaces with bounded Ricci curvature. These estimates are false on manifolds with only lower Ricci curvature bounds, and give the first taste of how analysis on a manifold with bounded Ricci curvature improves over that of a space with only lower Ricci curvature bounds.

Finally, in Section 8, we discuss the $4$ -dimensional case, and prove the finiteness up diffeomorphism theorem, Theorem 1.4. We also prove the improved $L^{2}$ curvature estimates of Theorem 1.5.

[01XL]

1.1. Outline of the proof Theorem 1.1, the codimension 4 conjecture

Let $S^{1}_{\beta}$ denote the circle of circumference $\beta<2\pi$ . It has been understood since [ChCo1] that to prove Theorem 1.1, the key step is to show that the cone $\mathds{R}^{n-2}\times C(S^{1}_{\beta})$ does not occur as the (pointed) Gromov-Hausdorff limit of some sequence $M^{n}_{j}$ with $|{\rm Ric}_{M^{n}_{j}}|\to 0$ . This was shown in [CCT02] assuming just a lower bound ${\rm Ric}_{M^{n}_{i}}\geq-(n-1)$ , but with the additional assumption that the $L^{1}$ norm of the curvature is sufficiently small. In [Ch2], it was proved for the Kähler-Einstein case, which was also done by Tian. A common feature of both of the proofs is an argument by contradiction, implemented by the use of harmonic almost splitting maps $u:B_{2}(p)\to\mathds{R}^{n-2}$ , see Lemma 1.7. In each case, it is shown that for most points $s\in\mathds{R}^{n-2}$ in the range, the slice $u^{-1}(s)$ has a certain good property which, when combined with the assumed curvature bounds, enables one to deduce a contradiction. In particular, in [CCT02] it is shown that most slices $u^{-1}(s)$ have integral bounds on the second fundamental form, which when combined with the assumed integral curvature bounds, enables one apply the Gauss-Bonnet formula for $2$ -dimensional manifolds with boundary, to derive a contradiction.

However, prior to the present paper it was not known how, in the general case, to implement a version of the above strategy which would rule out the cones $\mathds{R}^{n-2}\times C(S^{1}_{\beta})$ without assuming the integral curvature estimates. In the remainder of this subsection we will state the main results which are used in the present implementation and allow us to prove Theorem 1.1.

Thus, we consider a sequence of Riemannain manifolds $(M^{n}_{j},d_{j},p_{j})$ , with $|{\rm Ric}_{M^{n}_{j}}|\to 0$ and ${\rm Vol}(B_{1}(p_{j})>{\rm v}>0$ , such that

\displaystyle(M^{n}_{j},d_{j},p_{j})\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}\mathds{R}^{n-2}\times C(S^{1}_{\beta})\,.

(1.10)

We wish to see that $\beta=2\pi$ . As above, we have harmonic almost splitting maps

\displaystyle u_{j}:B_{2}(p_{j})\to\mathds{R}^{n-2}\,,

(1.11)

see Lemma 1.7 below. The key ingredient will be Theorem 1.8 (the Slicing Theorem), which states that there exist $s_{j}\in\mathds{R}^{n-2}$ such that for all $x\in u^{-1}_{j}(s_{j})$ and for all $r<1$ , the ball $B_{r}(x)$ is $\epsilon_{j}r$ -close in the Gromov-Hausdorff sense to a ball in an isometric product $\mathds{R}^{n-2}\times S_{j,x,r}$ , where $\epsilon_{j}\to 0$ as $j\to\infty$ .

Granted this, we can apply a blow up argument in the spirit of [A90] to obtain a contradiction. Namely, it is easy to see that if $\beta<2\pi$ then the minimum of the harmonic radius $r_{h}$ at points of the slice $u^{-1}_{j}(s_{j})$ is obtained at some $x_{j}\in u^{-1}_{j}(s_{j})$ and is going to zero as $j\to\infty$ . We rescale the metric by the inverse of the harmonic radius $r_{j}=r_{h}(x_{j})$ and find a subsequence converging in the pointed Gromov-Hausdorff sense a smooth noncompact Ricci flat manifold,

\displaystyle(M^{n}_{j},r_{j}^{-1}d_{j},x_{j})\to(X,d_{X},x)\,,

(1.12)

such that $X=\mathds{R}^{n-2}\times S$ splits off $\mathds{R}^{n-2}$ isometrically, with $S$ a smooth two dimensional surface. It follows that $S$ is Ricci flat, and hence flat. From the noncollapsing assumption, it follows that $X$ has Euclidean volume growth. Thus, $X=\mathds{R}^{n}$ is Euclidean space. However, the $2$ -sided Ricci bound implies that the harmonic radius behaves continuously in the limit. Hence, the harmonic radius at $x$ is $r_{h}(x_{\infty})=1$ ; a contradiction. See Section 5.1 for more details on the blow up argument.

Clearly then, the key issue is to show the existence of the points $s_{j}\in\mathds{R}^{n-2}$ , such that at all points $x\in u_{j}^{-1}(s_{j})$ , we have the above mentioned splitting property on $B_{r}(x)$ for all $r<1$ . To indicate the proof, we now recall some known connections between isometric splittings, the Gromov-Haudorff distance and harmonic maps to Euclidean spaces $\mathds{R}^{k}$ . We begin with a definition.

[01XM]

Definition 1.6.

A $\epsilon$ -splitting map $u=(u^{1},\ldots,u^{k}):B_{r}(p)\to\mathds{R}^{k}$ is a harmonic map such that:

(1)

$|\nabla u|\leq 1+\epsilon$ .
(2)

$\fint_{B_{r}(p)}|\langle\nabla u^{\alpha},\nabla u^{\beta}\rangle-\delta^{\alpha\beta}|^{2}<\epsilon^{2}$ .
(3)

$r^{2}\fint_{B_{r}(p)}|\nabla^{2}u^{\alpha}|^{2}<\epsilon^{2}$ .

Note that the condition that $u$ is harmonic is equivalent to the harmonicity of the individual component functions $u^{1},\dots,u^{k}$ .

The following lemma summarizes the basic facts about splitting maps²² 2 In [ChCo1], only a uniform bound $|\nabla u|<C(n)$ is proved. This would actually suffice for our present purposes. The improved bound, $|\nabla u|<1+\epsilon$ , in (1) above, is derived in (3.30)–(3.34), in a context that passes over almost verbatim to the present one.

[01XN]

Lemma 1.7 ([ChCo1]).

For every $\epsilon,R>0$ there exists $\delta=\delta(n,\epsilon,R)>0$ such that if ${\rm Ric}_{M^{n}}\geq-(n-1)\delta$ then:

(1)

If $u:B_{2R}(p)\to\mathds{R}^{k}$ is a $\delta$ -splitting map, then there exists a map $f:B_{R}(p)\to u^{-1}(0)$ such that

$(u,f):B_{R}(p)\to\mathds{R}^{k}\times u^{-1}(0)\,,$

is an $\epsilon$ -Gromov Hausdorff map, where $u^{-1}(0)$ is given the induced metric.
(2)

If

$\displaystyle d_{GH}(B_{\delta^{-1}}(p),B_{\delta^{-1}}(0))<\delta,$ (1.13)

where $0\in\mathds{R}^{k}\times Y$ , then there exists an $\epsilon$ -splitting map $u:B_{R}(p)\to\mathds{R}^{k}$ .

Let us return to the consideration of the maps $u_{j}$ from (1.11), which in our situation arise from (2) of Lemma 1.7. We can thus assume that the $u_{j}$ are $\delta_{j}$ -splitting maps, with $\delta_{j}\to 0$ . We wish to find slices $u^{-1}_{j}(s_{j})$ such that $B_{r}(x)$ continues to almost split for all $x\in u_{j}^{-1}(s_{j})$ and all $r\leq 1$ . One might hope that there always exist $s_{j}$ such that by restricting the map $u_{j}$ to each such ball $B_{r}(x)$ , one obtains an $\epsilon_{j}$ -splitting map. However, it turns out that there are counterexamples to this statement; see Example 2.1.

The essential realization is that for our purposes, it actually suffices to show the existence of $s_{j}$ such that for all $x\in u_{j}^{-1}(s_{j})$ and all $0<r\leq 1$ , there exists a matrix $A=A(x,r)\in GL(n-2)$ , such that the harmonic map $A\circ u_{j}:B_{r}(x)\to\mathds{R}^{n-2}$ is our desired $\epsilon_{j}$ -splitting map. Thus, while $u_{j}$ might not itself be a splitting map on $B_{r}(x)$ , it might only differ from one by a linear transformation of the image.³³ 3 Note that if such a matrix $A$ exists, without essential loss of generality, it can be chosen to be lower triangular. Since this condition also plays a role in the proof of Theorem 3.2, we will incorporate it from now on. This turns out to hold. More precisely, we have the following result.

[01XP]

Theorem 1.8.

(Slicing theorem) For each $\epsilon>0$ there exists $\delta(n,\epsilon)>0$ such that if $M^{n}$ satisfies ${\rm Ric}_{M^{n}}\geq-(n-1)\delta$ and if $u:B_{2}(p)\to\mathds{R}^{n-2}$ is a harmonic $\delta$ -splitting map, then there exists a subset $G_{\epsilon}\subseteq B_{1}(0^{n-2})$ which satisfies the following:

(1)

${\rm Vol}(G_{\epsilon})>{\rm Vol}(B_{1}(0^{n-2}))-\epsilon$ .
(2)

If $s\in G_{\epsilon}$ then $u^{-1}(s)$ is nonempty.
(3)

For each $x\in u^{-1}(G_{\epsilon})$ and $r\leq 1$ there exists a lower triangular matrix $A\in GL(n-2)$ such that $A\circ u:B_{r}(x)\to\mathds{R}^{n-2}$ is an $\epsilon$ -splitting map.

The proof of the Slicing Theorem is given in Section 4. We now describe main steps in the proof.

To begin with, by using Bochner’s formula and the improved Kato inequality, $|\nabla|\nabla u^{a}|\,|^{2}\leq\frac{n-1}{n}|\nabla^{2}u^{a}|^{2}$ , we show in Section 3.1 the following estimates on the ball $B_{2}(p)$ .

[01XQ]

Theorem 1.9.

(Higher order estimates) For every $\epsilon>0$ there exists $\delta(n,\epsilon)>0$ such that if ${\rm Ric}_{M^{n}}\geq-(n-1)\delta$ and $u:B_{2}(p)\to\mathds{R}^{k}$ is a $\delta$ -splitting map, then the following hold:

(1)

There exists $\alpha(n)>0$ such that for each $1\leq a\leq k$ ,

$\displaystyle\fint_{B_{1}(p)}\frac{|\nabla^{2}u^{a}|^{2}}{|\nabla u^{a}|^{1+\alpha}}<\epsilon\,.$ (1.14)
(2)

Let $\omega^{\ell}\equiv du^{1}\wedge\cdots\wedge du^{\ell}$ , $1\leq\ell\leq k$ . Then

$\displaystyle\fint_{B_{1}(p)}\big|\Delta|\omega^{\ell}|\big|<\epsilon\,.$ (1.15)

As will be clear from Theorem 1.11 below (the Transformation theorem) that the following definition is key.

[01XR]

Definition 1.10.

Let $u:B_{2}(p)\to\mathds{R}^{k}$ be a harmonic function and put $\omega^{\ell}=du^{1}\wedge\cdots\wedge du^{\ell}$ . For $x\in B_{1}(p)$ and $\delta>0$ , define the singular scale $s^{\delta}_{x}\geq 0$ to be the infimum of all radii $s$ such that for all $r$ with $s\leq r<\frac{1}{2}$ and all $1\leq\ell\leq k$ we have

\displaystyle r^{2}\fint_{B_{r}(x)}|\Delta|\omega^{\ell}||\leq\delta\fint_{B_{r}(x)}|\omega^{\ell}|\,.

(1.16)

Note that there is an invariance property for (1.16). Namely, if (1.16) holds for $u$ then it holds for $A\circ u$ for any lower triangular matrix $A\in GL(k)$ . That is, the singular scale of $u$ and the singular scale of $A\circ u$ are equal. In view of (1.15), this means essentially that (1.16) is a necessary condition for the existence of $A$ as in the Slicing theorem. Our next result, which is by far the most technically difficult of the paper, provides a sort of converse. We will not attempt to summarize the proof except to say that it involves a contradiction argument, as well as an induction on $\ell$ . It is proved in Section 3:

[01XS]

Theorem 1.11.

(Transformation theorem) For every $\epsilon>0$ there exists $\delta=\delta(n,\epsilon)>0$ such that if ${\rm Ric}_{M^{n}}\geq-(n-1)\delta^{2}$ and $u:B_{2}(p)\to\mathds{R}^{k}$ is a $\delta^{2}$ -splitting map, then for each $x\in B_{1}(p)$ and $r\geq s^{\delta}_{x}$ there exists a lower triangular matrix $A=A(x,r)$ such that $A\circ u:B_{r}(x)\to\mathds{R}^{k}$ is a $\epsilon$ -splitting map.

Granted the Transformation theorem, let us return to the outline of the proof of the Slicing theorem. So consider the singular radius $s_{x}^{\eta}$ , where $\eta(n,\epsilon)$ such that for $\delta<\eta^{2}$ the conclusions of Theorem 1.11 hold for $\epsilon>0$ . Let $u:B_{2}(p)\to\mathds{R}^{n-2}$ denote a harmonic $\delta$ -splitting map, and put

\displaystyle\mathcal{B}_{\eta}=:\,\bigcup_{x\,|\,s^{\delta}_{x}>0}B_{s^{\eta}_{x}}(x)\,.

(1.17)

Let $|u(B_{r}(x))|$ denote the $(n-2)$ -dimensional measure of the image $u(B_{r}(x))$ . In view of the Transformation theorem, to conclude the proof of the Slicing theorem it suffices to show

|u(\mathcal{B}_{\eta})|\leq\delta^{\prime}\,,

(1.18)

for $\delta^{\prime}<<\epsilon$ . To this end, we record two perhaps non-obvious, but easily verified consequences of Theorem 1.11.

Denote by $\mu$ , the measure such that for all open sets $U$

\mu(U)=\left(\int_{B_{1}(p)}|\omega|\right)^{-1}\cdot\int_{U}|\omega|\,.

The first consequence (see Lemma 4.1) is that for each $x$ and $1/2\geq r\geq s^{\eta}_{x}$ , we have the doubling condition

\mu(B_{2r}(x))\leq C(n)\cdot\mu(B_{r}(x))\,.

(1.19)

Let $|u(B_{r}(x)|$ denote the $(n-2)$ -dimensional measure of the image $u(B_{r}(x))$ .

The second consequence (see Lemma 4.2) is that if $1/2\geq r\geq s^{\eta}_{x}$ , then we have the volume estimate

|u(B_{r}(x)|\leq C(n)\cdot r^{-2}\mu(B_{r}(x))\,.

(1.20)

The proof of these results exploits the fact that $A\circ u:B_{r}(x)\to\mathds{R}^{n-2}$ is an $\epsilon$ -splitting map for some lower triangular matrix $A$ .

By a standard covering lemma, there exists a collection of mutually disjoint balls, $\{B_{s_{j}}(x_{j})\}$ with $s_{j}=s^{\eta}_{x_{j}}$ , such that

\mathcal{B}_{\eta}\subset\bigcup_{j}B_{6s_{j}}(x_{j})\,.

(1.21)

Since the balls $B_{s_{j}}(x_{j})$ are mutually disjoint, we can apply Theorem 1.9 together with (1.20) and the doubling property (1.19) of $\mu$ to obtain

$\displaystyle\|u(B_{\eta})\|$	$\displaystyle\leq\sum\|u(B_{6s_{j}}(x_{j}))\|\leq\sum({6s_{j}})^{-2}\mu(B_{6s_{j}}(x_{j}))$
	$\displaystyle\leq C(n)\sum s_{j}^{-2}\mu(B_{s_{j}}(x_{j}))\leq C\eta^{-1}\sum\int_{B_{s_{j}}}\|\Delta\|\omega^{\ell}\|\|$
	$\displaystyle\leq C\eta^{-1}\int_{B_{2}}\|\Delta\|\omega^{\ell}\|\,\|\leq\delta^{\prime}\,,$	(1.22)

where by Theorem 1.9 the last term tends to zero as $\delta\to 0$ , as claimed. See Section 4 for a complete proof of the Slicing Theorem.

[01XT]

2. Background and Preliminaries

In this section we review from standard constructions and techniques, which will be used throughout the paper.

[01XU]

2.1. Stratification of Limit Spaces

In this subsection we recall some basic properties of pointed Gromov-Hausdorff limit spaces

\displaystyle(M^{n}_{j},d_{j},p_{j})\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}(X,d,p)\,,

(2.1)

where the ${\rm Ric}_{M^{n}_{j}}\geq-(n-1)$ and the noncollapsing assumption ${\rm Vol}(B_{1}(p_{j}))\geq{\rm v}>0$ holds. In particular, we recall the stratification of a noncollapsed limit space, which was first introduced in [ChCo1], and which will play an important role in the proof of Theorem 1.1. The effective version, called the quantitative stratification, which was first introduced in [ChNa13], will be recalled in Section 7. It will play an important role in the estimates of Theorem 1.3.

Given $x\in X$ , we call a metric space $X_{x}$ a tangent cone at $x$ if there exists a sequence $r_{i}\to 0$ such that

\displaystyle(X,r_{i}^{-1}d,x)\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}X_{x}\,.

(2.2)

That tangent cones exist at every point is a consequence of Gromov’s compactness theorem; see for instance the book [P]. A point is called regular if every tangent cone is isometric to $\mathds{R}^{n}$ and otherwise singular. The set of singular points is denoted by $\mathcal{S}$ . As explained below, for noncollapsed limit spaces with a uniform lower Ricci bound, the singular set has codimension $\geq 2$ . At singular points, tangent cones may be highly nonunique with ill-defined dimension of the singular set, and even homeomorphism type, see for instance [CoNa2]. Easy examples show that the singular set need not be closed if one just assumes a uniform a lower bound ${\rm Ric}_{M^{n}_{j}}\geq-(n-1)$ . However, under the assumption of a $2$ -sided bound $|{\rm Ric}_{M^{n}_{j}}|\leq(n-1)$ , the singular set is indeed closed; see [A90], [ChCo2].

For noncollapsed limit spaces, as shown in [ChCo1], every tangent cone is a metric cone, i.e.

\displaystyle X_{x}=C(Z)\,,

(2.3)

for some compact metric space $Z$ , with ${\rm diam}(Z)\leq\pi$ . With this as our starting point, we introduce the following notion of symmetry.

[01XV]

Definition 2.1.

A metric space $Y$ is called $k$ -symmetric if $Y$ is isometric to $\mathds{R}^{k}\times C(Z)$ for some compact metric space $Z$ . We define the closed $k$ th-stratum by

\displaystyle\mathcal{S}^{k}(X)\equiv\{x\in X:\text{ no tangent cone at $x$ is $(k+1)$-symmetric}\}

(2.4)

Thus, in the noncollapsed case, every tangent cone is $0$ -symmetric.

The key result of [ChCo1] is the following:

\displaystyle\dim\mathcal{S}^{k}\leq k\,,

(2.5)

where dimension is in the Hausdorff sense. Thus, away from a set of dimension $k$ , every point has some tangent cone with $(k+1)$ degrees of symmetry. For an effective refinement of this theorem see [ChNa13] and Section 7.

[01XW]

2.2. $\epsilon$ -Regularity Theorems

A central result of this paper is the $\epsilon$ -regularity theorem, Theorem 6.1. The original $\epsilon$ -regularity theorems for Einstein manifolds were given in [A90], [T90] , [BKN89]. They state that if $M^{n}$ is an Einstein manifold, ${\rm Ric}_{M^{n}}=\lambda g$ , with $|\lambda|\leq n-1$ , and if for $B_{2}(p)\subset M^{n}$ ,

\displaystyle\fint_{B_{2}(p)}|{\rm Rm}|^{n/2}<\epsilon(n)\,,

(2.6)

then $\sup_{B_{1}(p)}|{\rm Rm}|\leq 1$ .

In [CCT02], [Ch2], [CD13], $\epsilon$ -regularity theorems were proved under the assumption of $L^{q}$ curvature bounds, $1\leq q<n/2$ , provided $B_{2}(p)$ is assumed sufficiently close to a ball in a cone which splits off an isometric factor $\mathds{R}^{n-2q}$ .

On the other hand, the regularity theory of [ChNa13] for Einstein manifolds depends on $\epsilon$ -regularity theorems which do not assume any $L^{q}$ curvature bounds. In particular, it follows from the work of [A90] that there exists $\epsilon(n)>0$ such that if $|{\rm Ric}_{M^{n}}|\leq\epsilon(n)$ and if

\displaystyle d_{GH}(B_{2}(p),B_{2}(0^{n}))<\epsilon(n)\,,

(2.7)

where $B_{2}(0^{n})\subseteq\mathbb{R}^{n}$ , then $|{\rm Rm}|\leq 1$ on $B_{1}(x)$ .

This result can be extended in several directions. In order to state the extension in full generality, we first recall the notion of the harmonic radius:

[01XX]

Definition 2.2.

For $x\in X$ , we define the harmonic radius $r_{h}(x)$ so that $r_{h}(x)=0$ if no neighborhood of $x$ is a Riemannian manifold. Otherwise we define $r_{h}(x)$ to be the largest $r>0$ such that there exists a mapping $\Phi:B_{r}(0^{n})\to X$ such that:

(1)

$\Phi(0)=x$ with $\Phi$ is a diffeomorphism onto its image.
(2)

$\Delta_{g}x^{\ell}=0$ , where $x^{\ell}$ are the coordinate functions and $\Delta_{g}$ is the Laplace Beltrami operator.

(3)

If $g_{ij}=\Phi^{*}g$ is the pullback metric, then

\displaystyle||g_{ij}-\delta_{ij}||_{C^{0}(B_{r}(0^{n}))}+r||\partial_{k}g_{ij}||_{C^{0}(B_{r}(0^{n}))}\leq 10^{-3}\,.

(2.8)

We call a mapping $\Phi:B_{r}(0^{n})\to X$ as above a harmonic coordinate system. Harmonic coordinates have an abundance of good properties when it comes to regularity issues; see the book [P] for a nice introduction. In particular, if the Ricci curvature is uniformly bounded then in harmonic coordinates, the metric, $g_{ij}$ has a priori $C^{1,\alpha}\cap W^{2,q}$ bounds, for all $\alpha<1$ and $q<\infty$ . If in addition, there is a bound on $|\nabla{\rm Ric}_{M^{n}}|$ , then in harmony coordinates, $g_{ij}$ has $C^{2,\alpha}$ bounds, for all $\alpha<1$ .

The primary theorem we wish to review in this subsection is the following:

[01XY]

Theorem 2.3 ([A90], [ChCo1]).

There exists $\epsilon(n,{\rm v})>0$ such that if $M^{n}$ satisfies $|{\rm Ric}_{M^{n}}|\leq\epsilon$ , ${\rm Vol}(B_{1}(p))>{\rm v}>0$ , and

\displaystyle d_{GH}(B_{2}(p),B_{2}(0))<\epsilon(n)\,,

(2.9)

where $0\in\mathds{R}^{n-1}\times C(Z)$ , then the harmonic radius $r_{h}(x)$ satisfies

\displaystyle r_{h}(x)\geq 1\,.

(2.10)

If $M^{n}$ is further assumed to be Einstein, then the regularity scale $r_{x}$ satisfies $r_{x}\geq 1$ .

By the results of the previous subsection, it is possible to find balls satisfying the above constraint off a subset of Hausdorff codimension $2$ . Moreover, when combined with the quantitative stratification of [ChNa13], see also Section 7, this $\epsilon$ -regularity theorem leads to a priori $L^{p}$ bounds on the curvature. The primary result of the present paper can be viewed as Theorem 6.1, which states that the conclusions of Theorem 2.3 continue to hold if $\mathds{R}^{n-1}$ is replaced by $0\in\mathds{R}^{n-3}\times C(Z)$ .

[01XZ]

2.3. Examples

In this subsection, we indicate some simple examples which play an important role in guiding the results of this paper.

[01Y0]

Example 2.1.

(The Cone Space $\mathds{R}^{n-2}\times C(S^{1}_{\beta})$ ) The main result of this paper, Theorem 1.1, states that $\mathds{R}^{n-2}\times C(S^{1}_{\beta})$ , with $\beta<2\pi$ , is not the noncollapsed Gromov-Hausdorff limit of a sequence of manifolds with bounded Ricci curvature. However, it is clear that this space is the Gromov-Hausdorff limit of a a sequence of noncollapsed manifolds with a uniform lower Ricci curvature bound. Indeed, by rounding off $C(S^{1}_{\beta})$ we see that $\mathds{R}^{n-2}\times C(S^{1}_{\beta})$ can appear as a noncollapsed limit of manifolds with nonnegative sectional curvature.

In this example, let us just consider the two dimensional cone $C(S^{1}_{\beta})$ with $\beta<2\pi$ . Regard $S^{1}_{\beta}$ as $0\leq\theta\leq 2\pi$ , with the end points identified. Then the Laplacian on $S^{1}_{\beta}$ is $(\frac{2\pi}{\beta})^{2}\cdot\frac{\partial^{2}}{\partial\theta^{2}}$ . The eigenfunctions are of the form $e^{ik\theta}$ , where $k$ is an integer. Written in polar coordinates, a basis for the bounded harmonic functions on $C(S^{1}_{\beta})$ is $\{r^{\frac{2\pi}{\beta}|k|}\cdot e^{ik\theta}\}$ . In particular, we see from this that if $\beta<2\pi$ then $|\nabla(r^{\frac{2\pi}{\beta}|k|}\cdot e^{ik\theta})|\to 0$ as $r\to 0$ . As a consequence, every bounded harmonic function has vanishing gradient at the vertex, which is a set of positive $(n-2)$ -dimensional Hausdorff measure. By considering examples with more vertices, we can construct limit spaces where bounded harmonic functions $h$ must have vanishing gradient on bounded subsets sets of arbitrarily large, or even infinite, $(n-2)$ -dimensional Hausdorff measure. This set can even be taken to be dense.

[01Y1]

Example 2.2.

(The Eguchi-Hanson manifold) The Eguchi-Hanson metric $g$ is a complete Ricci flat metric on the cotangent bundle of $S^{2}$ , which at infinity, becomes rapidly asymptotic to the metric cone on $\mathds{R}\mathds{P}(3)$ or equivalently to $\mathds{R}^{4}/\mathds{Z}_{2}$ , where $\mathds{Z}_{2}$ acts on $\mathds{R}^{4}$ by $x\to-x$ . When the metric $g$ is scaled down by $g\to r^{2}g$ , with $r\to 0$ , one obtains a family of Ricci flat manifolds whose Gromov-Hausdorff limit is $C(\mathds{R}\mathds{P}(3))=\mathds{R}^{4}/\mathds{Z}_{2}$ . This is the simplest example which shows that even under the assumption of Ricci flatness and noncollapsing, Gromov-Hausdorff limit spaces can contain codimension 4 singularities.

[01Y2]

Example 2.3.

(Infinitely many topological types in dimension 4) Let $T^{3}$ denote a flat $3$ -torus. According to Anderson [A93], there is a collapsing sequence of manifolds $(M^{4}_{j},d_{j})\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}T^{3}$ satisfying

		$\displaystyle{\rm diam}(M^{4}_{j})\leq 1\,,$
		$\displaystyle\|{\rm Ric}_{M^{n}_{j}}\|\leq\epsilon_{j}\to 0\,,$
		$\displaystyle{\rm Vol}(M^{4}_{j})\to 0\,,$
		$\displaystyle b_{2}(M^{4}_{j})\to\infty\,,$		(2.11)

where $b_{2}(M^{4}_{j})$ denotes the second Betti number of $M^{4}_{j}$ . In particular, Theorem 1.4 , the finiteness theorem in dimension $4$ , does not extend to the case in which the lower volume bound is dropped.

[01Y3]

3. Proof of the Transformation Theorem

In this section we prove the Transformation theorem (Theorem 1.11) which is the main technical tool in the proof of the Slicing theorem (Theorem 1.8). As motivation, we mention the following. Given $\epsilon,\eta>0$ and a $\delta(\epsilon,\eta)$ -splitting map $u:B_{2}(p)\to\mathds{R}^{k}$ , one can use a weighted maximal function estimate for $|\nabla^{2}u|$ to conclude that away from a set $B$ of small $(n-2+\eta)$ -content that the restriction $u:B_{r}(x)\to\mathds{R}^{k}$ is an $\epsilon$ -splitting map for all $0<r<1$ . However, as we have observed in Example 2.1, we cannot take $\eta=0$ , since $|\nabla u|$ can vanish on a set of large $(n-2)$ -content. For purposes of proving the Slicing theorem, this set is too large.

Suppose instead, that we consider the collection of balls $B_{r}(x)$ such that for no lower triangular matrix $A\in GL(n-2)$ is $A\circ u$ an $\epsilon$ -splitting map on $B_{r}(x)$ . Though we cannot show that this set has small $(n-2)$ -content, we will prove that its image under $u$ has small $(n-2)$ -dimensional measure. This will be what is required for the Slicing Theorem.

For the case of a single function, $k=1$ , the basic idea can be explained as follows. In order to obtain an $\epsilon$ -splitting function on $B_{r}(x)$ , it is not necessary that the Hessian of $u$ is small and the gradient is close to $1$ or even has a definite lower bound. Rather, we need only that the Hessian of $u$ is small relative to the gradient. That is, consider the condition

r\fint_{B_{2r}(x)}|\nabla^{2}u|\leq\delta(\epsilon)\cdot\fint_{B_{2r}(x)}|\nabla u|\,.

(3.1)

If $\fint_{B_{2r}(x)}|\nabla u|$ is very small, then the restricted map $u:B_{r}(x)\to\mathds{R}$ will not define a splitting map. However, we may simply rescale $u$ so that $\fint_{B_{2r}(x)}|\nabla u|=1$ , in which case standard arguments as in the proof of Lemma 1.7 tell us that after the rescaling $u:B_{r}(x)\to\mathds{R}$ becomes an $\epsilon$ -splitting.

To control the collection of balls which do not satisfy the inequality (3.1), we start with integral estimate

\fint_{B_{3/2}(p)}\frac{|\nabla^{2}u|^{2}}{|\nabla u|}<\delta^{2}\,.

(3.2)

This will enable us to control the set of balls $B_{2r}(x)$ which do not satisfy

r^{2}\fint_{B_{2r}(x)}\frac{|\nabla^{2}u|^{2}}{|\nabla u|}<\delta\fint_{B_{2r}(x)}|\nabla u|\qquad({\rm for\,\,all}\,\,r)

(3.3)

and in particular, to show in Section 4 that the image under $u$ of this collection of balls has small $(n-2)$ -dimensional measure. On the other hand, (3.3) implies (3.1), since

r\fint_{B_{2r}(x)}|\nabla^{2}u|\leq\Big(r^{2}\fint_{B_{2r}(x)}\frac{|\nabla^{2}u|^{2}}{|\nabla u|}\Big)^{1/2}\cdot\Big(\fint_{B_{2r}(x)}|\nabla u|\Big)^{1/2}\leq C\delta^{1/2}\fint_{B_{2r}(x)}|\nabla u|\,.

(3.4)

For the case $k>1$ , two serious new issues arise. For one thing, even if on some ball $B_{r}(x)$ the gradients, $\nabla u^{1},\ldots,\nabla u^{n-2}$ satisfy (3.1) and we then normalize them to have $L^{2}$ norm $1$ , it still might be the case that in the $L^{2}$ sense, this normalized collection looks close to being linearly dependent. Then $u$ would be far from defining an $\epsilon$ -splitting map. The purpose of the Transformation theorem is to deal appropriately with this issue. Unfortunately, for $k>1$ , we are unable to obtain a precise analog of (3.2), which was the tool for handling the case $k=1$ . Instead, we have to proceed on the basis of (3.10), which makes the proof of the Transformation theorem in the general case substantially more difficult.

[01Y4]

3.1. Higher Order Estimates

A central analytic estimate from [ChCo1] in the proof of Lemma 1.7 states that if ${\rm Ric}_{M^{n}}\geq-(n-1)\delta$ and if $u:B_{2}(p)\to\mathds{R}$ is a harmonic function, then on $B_{1}(p)$ we have the $L^{2}$ estimates of the form

\fint_{B_{r}(x)}|\nabla^{2}u|^{2}\leq C(n)\inf_{c}\fint_{B_{2r}(x)}\Big|\,|\nabla u|^{2}-c\Big|+\delta\fint_{B_{2r}(x)}|\nabla u|^{2}\,.

(3.5)

Since the technique of proof will occur repeatedly in the sequel, we will recall it here.

According to [ChCo1] if ${\rm Ric}_{M^{n}}\geq-\delta$ , for any $B_{r}(x)\subset M^{n}$ there exists a cutoff function, with $0\leq\varphi\leq 1$ , such that

	$\displaystyle\varphi(x)$	$\displaystyle\equiv 1\text{ if }x\in B_{9r/5}(x)\,,$		(3.6)
	$\displaystyle{\rm supp}\,\varphi$	$\displaystyle\subset B_{2r}(x)\,,$		(3.6)

and such that

	$\displaystyle r\|\nabla\varphi\|$	$\displaystyle\leq C(n)\,,$		(3.7)
	$\displaystyle r^{2}\|\Delta\varphi\|$	$\displaystyle\leq C(n)\,.$		(3.7)

Now using Bochner’s formula, we get for $\varphi$ as above and any constant $c$ ,

$\displaystyle C(n)\fint_{B_{2r}(x)}\|\,\|\nabla u\|^{2}-c\|$	$\displaystyle\geq\fint_{B_{2r}(x)}\|\Delta\varphi\|\cdot\|\nabla u\|^{2}-c\|$	(3.8)
	$\displaystyle\geq\left\|\fint_{B_{2r}(x)}\varphi\Delta(\|\nabla u\|^{2}-c)\right\|$
	$\displaystyle=\left\|\fint_{B_{2}(x)}\varphi(\|\nabla^{2}u\|^{2}+{\rm Ric}(\nabla u,\nabla u))\right\|$
	$\displaystyle\geq\fint_{B_{r}(x)}\|\nabla^{2}u\|^{2}-\delta\|\nabla u\|^{2}\,,$

which implies (3.5).

Part (1) of the following Theorem 1.9, whose statement is recalled below, is actually a sharpening of (3.5).

For every $\epsilon>0$ there exists $\delta(n,\epsilon)>0$ such that if ${\rm Ric}_{M^{n}}\geq-\delta$ with $u:B_{2}(p)\to\mathds{R}^{k}$ a $\delta$ -splitting map, then the following hold:

(1)

There exists $\alpha(n)>0$ such that for each $1\leq a\leq k$ ,

$\displaystyle\fint_{B_{3/2}(p)}\frac{|\nabla^{2}u^{a}|^{2}}{|\nabla u^{a}|^{1+\alpha}}<\epsilon\,.$ (3.9)
(2)

Let $\omega^{\ell}\equiv du^{1}\wedge\cdots\wedge du^{\ell}$ , $1\leq\ell\leq k$ . Then

$\displaystyle\fint_{B_{3/2}(p)}\big|\Delta|\omega^{\ell}|\big|<\epsilon\,.$ (3.10)

[01Y5]

Proof of Theorem 1.9.

We begin with the proof of (3.9).

The Bochner formula for $|\nabla u|^{1-\alpha}$ is given by

\displaystyle\Delta|\nabla u|^{1-\alpha}=(1-\alpha)\frac{\Big(|\nabla^{2}u|^{2}-(1+\alpha)|\nabla|\nabla u||^{2}+{\rm Ric}(\nabla u,\nabla u)\Big)}{|\nabla u|^{1+\alpha}}\,.

(3.11)

To estimate the left hand side, we observe that since ${\rm trace}(\nabla^{2}u)=\Delta u=0$ , if follows if $\lambda_{1},\ldots,\lambda_{n}$ are the eigenvalues of $\nabla^{2}u$ then $\sum\lambda_{i}=0$ . In particular, if $\lambda_{n}$ is the largest eigenvalue then by the Schwarz inequality,

\displaystyle\lambda_{1}^{2}+\cdots+\lambda^{2}_{n}\geq\frac{1}{n-1}(\lambda_{1}+\cdots+\lambda_{n-1})^{2}+\lambda_{n}^{2}\geq\frac{n}{n-1}\lambda_{n}^{2}\,.

(3.12)

This leads to the improved Kato inequality

\displaystyle|\nabla^{2}u({\rm v})|^{2}\leq\big(1-\frac{1}{n}\big)|\nabla^{2}u|^{2}\,,

(3.13)

where ${\rm v}$ is any vector with $|{\rm v}|=1$ .

If rewrite

\displaystyle|\nabla|\nabla u||=\big|\nabla^{2}u\Big(\frac{\nabla u}{\nabla u}\Big)\big|\,,

(3.14)

and apply the improved Kato inequality, then we get the Bochner formula

\displaystyle\Delta|\nabla u|^{1-\alpha}\geq\frac{1-(n-1)\alpha}{n}\frac{|\nabla^{2}u|^{2}}{|\nabla u|^{1+\alpha}}-(1-\alpha)\delta|\nabla u|^{1-\alpha}\,.

(3.15)

which gives nontrivial information for any $\alpha<\frac{1}{n-1}$ . Namely,

\displaystyle\frac{|\nabla^{2}u|^{2}}{|\nabla u|^{1+\alpha}}\leq C(n,\alpha)\Big(\Delta|\nabla u|^{1-\alpha}+\delta|\nabla u|^{1-\alpha}\Big)\,.

(3.16)

As in the proof of (3.5), let $\varphi\geq 0$ be a smooth function as in [ChCo1], with ${\rm supp}\,\varphi\subset B_{2}(p)$ , $|\nabla\phi|,|\Delta\phi|\leq C(n)$ .

By multiplying both sides of (3.16) by $\varphi$ and integrating we obtain

$\displaystyle\int\varphi\frac{\|\nabla^{2}u\|^{2}}{\|\nabla u\|^{1+\alpha}}$	$\displaystyle\leq C(n,\alpha)\int\Big(\varphi\Delta\|\nabla u\|^{1-\alpha}+\varphi\delta\|\nabla u\|^{1-\alpha}\Big)\,,$
	$\displaystyle\leq C(n,\alpha)\int\Delta\varphi\Big(\|\nabla u\|^{1-\alpha}-\fint_{B_{2}(p)}\|\nabla u\|^{1-\alpha}\Big)+C(n,\alpha)\delta\int_{B_{2}(p)}\|\nabla u\|^{1-\alpha}\,,$
	$\displaystyle\leq C(n,\alpha)\int_{B_{2}(p)}\Big\|\|\nabla u\|^{1-\alpha}-\fint_{B_{2}(p)}\|\nabla u\|^{1-\alpha}\Big\|+C(n,\alpha)\delta\int_{B_{2}(p)}\|\nabla u\|^{1-\alpha}\,,$	(3.17)

Now we use that if $u$ is a harmonic $\delta$ -splitting map then $|\nabla u|^{1-\alpha}$ is bounded and $\fint_{B_{2}(p)}\Big||\nabla u|^{1-\alpha}-\fint_{B_{2}(p)}|\nabla u|^{1-\alpha}\Big|$ is small. In particular, for $\delta$ sufficiently small, we have

	$\displaystyle\fint_{B_{3/2}(p)}\frac{\|\nabla^{2}u\|^{2}}{\|\nabla u\|^{1+\alpha}}$	$\displaystyle\leq C(n)\fint_{B_{2}(p)}\varphi\frac{\|\nabla^{2}u\|^{2}}{\|\nabla u\|^{1+\alpha}}$
		$\displaystyle\leq C(n,\alpha)\fint_{B_{2}(p)}\Big\|\|\nabla u\|^{1-\alpha}-\fint_{B_{2}(p)}\|\nabla u\|^{1-\alpha}\Big\|+C(n,\alpha)\delta\fint_{B_{2}(p)}\|\nabla u\|^{1-\alpha}\leq\epsilon\,,$		(3.18)

which proves (3.9).

Now we proceed with the proof of (3.10).

We begin with some computations. Given $\omega^{\ell}=du^{1}\wedge\cdots\wedge du^{\ell}$ we have

		$\displaystyle\Delta\omega^{\ell}=\sum_{b}du^{1}\wedge\cdots{\rm Ric}(du^{a})\wedge\cdots\wedge du^{\ell}+\sum_{a,b}du^{1}\wedge\nabla^{j}(du^{a})\wedge\cdots\wedge\nabla_{j}(du^{b})\wedge\cdots\wedge du^{\ell}\,,$
		$\displaystyle\Delta\|\omega^{\ell}\|=\frac{\|\nabla\omega^{\ell}\|^{2}-\|\nabla\|\omega^{\ell}\|\|^{2}}{\|\omega^{\ell}\|}+\langle\sum_{b}du^{1}\wedge\cdots{\rm Ric}(du^{a})\wedge\cdots\wedge du^{\ell},\frac{\omega^{\ell}}{\|\omega^{\ell}\|}\rangle$
		$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,+\langle\sum_{a,b}du^{1}\wedge\nabla^{j}(du^{a})\wedge\cdots\wedge\nabla_{j}(du^{b})\wedge\cdots\wedge du^{\ell},\frac{\omega^{\ell}}{\|\omega^{\ell}\|}\rangle\,.$		(3.19)

In particular, if $u$ is a $\delta$ -splitting map then

\displaystyle\Delta|\omega^{\ell}|-\langle\sum_{a,b}du^{1}\wedge\nabla^{j}(du^{a})\wedge\cdots\wedge\nabla_{j}(du^{b})\wedge\cdots\wedge du^{\ell},\frac{\omega^{\ell}}{|\omega^{\ell}|}\rangle+C(n)\delta|\omega^{\ell}|\geq 0\,,

(3.20)

As in the proof of (3.5), let $\varphi\geq 0$ be a smooth function as in [ChCo1], with ${\rm supp}\,\varphi\subset B_{2}(p)$ , $|\nabla\varphi|,|\Delta\phi|\leq C(n)$ . Then we have

$\displaystyle\fint_{B_{2}(p)}\varphi\Big\|$	$\displaystyle\Delta\|\omega^{\ell}\|-\langle\sum_{a,b}du_{a_{1}}\wedge\nabla^{j}(du_{a_{a}})\wedge\cdots\wedge\nabla_{j}(du_{a_{b}})\wedge\cdots\wedge du_{a_{k}},\frac{\omega^{\ell}}{\|\omega^{\ell}\|}\rangle+C(n)\delta\|\omega^{\ell}\|\Big\|$
	$\displaystyle=\fint_{B_{2}(p)}\varphi\Big(\Delta\|\omega^{\ell}\|-\langle\sum_{a,b}du_{a_{1}}\wedge\nabla^{j}(du_{a_{a}})\wedge\cdots\wedge\nabla_{j}(du_{a_{b}})\wedge\cdots\wedge du_{a_{k}},\frac{\omega^{\ell}}{\|\omega^{\ell}\|}\rangle+C(n)\delta\|\omega^{\ell}\|\Big)$
	$\displaystyle\leq\fint_{B_{2}(p)}\Delta\varphi\Big(\|\omega^{\ell}\|-1\Big)+\sum\fint_{B_{2}(p)}\|\nabla^{2}u_{a_{j}}\|^{2}+C\delta\fint_{B_{2}(p)}\|\omega^{\ell}\|\,,$
	$\displaystyle\leq C(n)\fint\big\|\|\omega^{\ell}\|-1\big\|+C\delta<\frac{\epsilon}{3}\,,$	(3.21)

if $\delta(n,\epsilon)$ is sufficiently small. Thus, if $\delta=\delta(n,\epsilon)$ is sufficiently small,

$\displaystyle\fint_{B_{2}(p)}$	$\displaystyle\varphi\Big\|\Delta\|\omega^{\ell}\|\Big\|\leq\fint_{B_{2}(p)}\phi\Big\|\Delta\|\omega^{\ell}\|-\langle\sum_{a,b}du^{1}\wedge\nabla^{j}(du^{a})\wedge\cdots\wedge\nabla_{j}(du^{b})\wedge\cdots\wedge du^{\ell},\frac{\omega^{\ell}}{\|\omega^{\ell}\|}\rangle+C(n)\delta\|\omega^{\ell}\|\Big\|$
	$\displaystyle+\fint_{B_{2}(p)}\varphi\Big\|\langle\sum_{a,b}du^{1}\wedge\nabla^{c}(du^{a})\wedge\cdots\wedge\nabla_{c}(du^{b})\wedge\cdots\wedge du^{\ell},\frac{\omega^{\ell}}{\|\omega^{\ell}\|}\rangle\Big\|+C(n)\delta\fint_{B_{2}(p)}\varphi\|\omega^{\ell}\|\,,$
	$\displaystyle\leq\frac{\epsilon}{3}+C\sum\fint_{B_{2}(p)}\|\nabla^{2}u^{a}\|^{2}+C\delta\fint_{B_{2}(p)}\|\omega^{\ell}\|\,.$
	$\displaystyle\leq\epsilon\,,$	(3.22)

This completes the proof of (3.10). ∎

[01Y6]

3.2. Proof of the Transformation theorem

In this subsection, we prove the Transformation theorem (Theorem 1.11) which constitutes the technical heart of the Slicing theorem (Theorem 1.8). We will assume for notational simplicity that $M^{n}$ is complete, but it is an easy exercise to show that this may be weakened to the local assumption that $B_{4}(p)$ has compact closure in $M^{n}$ . First we recall the definition of the singular scale:

Let $u:B_{2}(p)\to\mathds{R}^{k}$ be a harmonic function. For $\delta>0$ let us define for $x\in B_{1}(p)$ the singular scale $s^{\delta}_{x}\geq 0$ as the infimum of all radii $s$ , such that for all $s<r<\frac{1}{2}$ and all $1\leq\ell\leq k$ we have the estimate

r^{2}\fint_{B_{r}(x)}|\Delta|\omega^{\ell}||\leq\delta\fint_{B_{r}(x)}|\omega^{\ell}|\,,

where $\omega^{\ell}=du^{1}\wedge\cdots\wedge du^{\ell}$ .

Next recall that Theorem 1.11 states the following.

For every $\epsilon>0$ there exists $\delta=\delta(n,\epsilon)>0$ such that if ${\rm Ric}_{M^{n}}\geq-\delta^{2}$ and $u:B_{2}(p)\to\mathds{R}^{k}$ is a harmonic $\delta^{2}$ -splitting map, then for each $x\in B_{1}(p)$ and $r\geq s^{\delta}_{x}$ there exists a lower triangular matrix $A=A(x,r)$ such that $A\circ u:B_{r}(x)\to\mathds{R}^{k}$ is a harmonic $\epsilon$ -splitting map.

[01Y7]

Proof of Theorem 1.11:

The strategy will be a proof by induction. Thus, we will begin with the simplest case of $k=1$ . The following is a slightly more general form of the statement we wish to prove.

[01Y8]

Lemma 3.1.

Let $u:B_{2r}(x)\to\mathds{R}$ be a harmonic function with $r\leq 1$ . Then for every $\epsilon>0$ there exists $\delta(n,\epsilon)>0$ such that if ${\rm Ric}_{M^{n}}\geq-(n-1)\delta^{2}$ and

\displaystyle r^{2}\fint_{B_{2r}(x)}|\Delta|\nabla u||\leq\delta\fint_{B_{2r}(x)}|\nabla u|\,,

(3.23)

then for $A=\Big(\fint_{B_{r}(x)}|\nabla u|\Big)^{-1}>0$ we have that $A\circ u:B_{r}(x)\to\mathds{R}$ is an $\epsilon$ -splitting map.

As in (3.6)–(3.8), the Bochner formula (3.11) and the fact that $u$ is harmonic leads to the improved Kato inequality, $|\nabla|\nabla u^{a}|\,|^{2}\leq\frac{n-1}{n}|\nabla^{2}u^{a}|^{2}$ , from which we can compute

\displaystyle\Delta|\nabla u|\geq\frac{1}{n}\frac{|\nabla^{2}u|^{2}}{|\nabla u|}-(n-1)\delta^{2}|\nabla u|\,.

(3.24)

In particular, the estimate (3.23) gives rise to the estimate

\displaystyle r^{2}\fint_{B_{2r}(x)}\frac{|\nabla^{2}u|^{2}}{|\nabla u|}\leq C(n)\delta\fint_{B_{2r}(x)}|\nabla u|\,,

(3.25)

from which, as previously noted (see (3.3), (3.4) ) we get

\displaystyle r\fint_{B_{2r}(x)}|\nabla^{2}u|\leq\Big(r^{2}\fint_{B_{2r}(x)}\frac{|\nabla^{2}u|^{2}}{|\nabla u|}\Big)^{1/2}\cdot\Big(\fint_{B_{2r}(x)}|\nabla u|\Big)^{1/2}\leq C\delta^{1/2}\fint_{B_{2r}(x)}|\nabla u|\,.

(3.26)

Let us put $v=\Big(\fint_{B_{2r}(x)}|\nabla u|\Big)^{-1}u$ , so that $\fint_{B_{2r}(x)}|\nabla v|=1$ . The lower Ricci bound implies that a Poincaré inequality holds. When combined with the last inequality this implies

\displaystyle\fint_{B_{2r}(x)}\big||\nabla v|-1\big|\leq C(n)\delta^{1/2}\,.

(3.27)

By using the doubling property, we have after possible increasing $C(n)$ , that for every $y\in B_{3r/2}(x)$

\displaystyle\fint_{B_{r/2}(y)}\big||\nabla v|-1\big|\leq C\delta^{1/2}\,.

(3.28)

In particular,

\displaystyle 1-C\delta^{1/2}\leq\frac{\fint_{B_{2r}(x)}|\nabla u|}{\fint_{B_{r}(x)}|\nabla u|}\leq 1+C\delta^{1/2}\,.

(3.29)

Hence, if we can show that, $\delta$ sufficiently small, the map $v:B_{r}(x)\to\mathds{R}$ is an $\epsilon/2$ -splitting, for $k=1$ , the proof will be complete

Now as in [ChCo1], let $\varphi\geq 0$ be a cutoff function satisfying $\varphi(y)=1$ if $y\in B_{5r/3}(x)$ with $\varphi(y)\equiv 0$ if $y\not\in B_{2r}(x)$ , and such that $r|\nabla\varphi|,r^{2}|\Delta\varphi|\leq C(n)$ . Let $\rho_{t}(y,dz)$ be the heat kernel on $M^{n}$ . Consider for $y\in B_{3r/2}(x)$ the one parameter family

\displaystyle\int\big(|\nabla v|-1\big)\phi\,\rho_{t}(y,dz)\,.

(3.30)

Note that

$\displaystyle\frac{d}{dt}\int\Big(\|\nabla v\|-1\Big)\varphi\,\rho_{t}(y,dz)$	$\displaystyle=\int\Big(\frac{\|\nabla^{2}v\|}{\|\nabla v\|}\varphi+\langle\nabla\|\nabla v\|,\nabla\varphi\rangle+(\|\nabla v\|-1)\Delta\varphi\Big)\rho_{t}(y,dz)\,,$
	$\displaystyle\geq-C(n)\int_{A(3r/2,2r)}r^{-1}\|\nabla^{2}v\|+r^{-2}\big\|\|\nabla v\|-1\big\|\rho_{t}(y,dz)\,,$
	$\displaystyle\geq-C\delta^{1/2}r^{-2}\,,$	(3.31)

where the last inequality is for $t\in[0,r^{2}]$ . Integrating this yields

\displaystyle(|\nabla v|(y)-1)\leq C\delta^{1/2}+\int\Big(|\nabla v|-1\big)\varphi\,\rho_{r^{2}}(y,dz)\leq C\delta^{1/2}+\fint_{B_{2r}(x)}\big||\nabla v|-1\big|\leq C\delta^{1/2}\,.

(3.32)

In particular we have

\displaystyle\sup_{B_{3r/2}(x)}|\nabla v|\leq 1+C\delta^{1/2}\,.

(3.33)

Combining this with the integral estimate (3.28) we get

\displaystyle\fint_{B_{3r/2}(x)}\big||\nabla v|^{2}-1\big|\leq C\delta^{1/2}\,.

(3.34)

Now using the Bochner formula

\displaystyle\Delta|\nabla v|^{2}=2|\nabla^{2}v|^{2}+2{\rm Ric}(\nabla v,\nabla v)\geq 2|\nabla^{2}v|^{2}-C\delta^{2}|\nabla v|^{2}\,,

(3.35)

we can estimate

$\displaystyle\fint_{B_{r}(x)}\|\nabla^{2}v\|^{2}$	$\displaystyle\leq C(n)\fint_{B_{3r/2}(x)}\varphi\|\nabla^{2}v\|^{2}$	(3.36)
	$\displaystyle\leq C\fint_{B_{3r/2}(x)}\varphi\Big(\Delta(\|\nabla v\|^{2}-1)+\delta^{2}\|\nabla v\|^{2}\Big)$
	$\displaystyle\leq C\fint_{B_{3r/2}(x)}\|\Delta\varphi\|\big\|\|\nabla v\|^{2}-1\big\|+C\delta^{2}\fint_{B_{3r/2}(x)}\|\nabla v\|^{2}\,,$
	$\displaystyle\leq Cr^{-2}\delta^{1/2}\,.$	(3.37)

Hence, for $\delta(n,\epsilon)$ sufficiently we have that $v$ is an $\epsilon/2$ -splitting, which as previously remarked, proves the theorem for the case $k=1$ .

We now turn to the proof of Theorem 1.11, which will proceed by induction.

Assume the Theorem has been proved for some $k-1\geq 1$ . We will prove the result for $k$ by arguing by contradiction.

Thus, we can suppose that for some $\epsilon>0$ the result is false. There is no harm is assuming $0<\epsilon\leq\epsilon(n)$ is sufficiently small. Then, for some $\delta_{j}\to 0$ we can find a sequence of spaces $(M^{n}_{j},g_{j},p_{j})$ with ${\rm Ric}_{M^{n}_{j}}\geq-\delta_{j}^{2}$ and mappings $u_{j}:B_{2}(p_{j})\to\mathds{R}^{k}$ which are $\delta_{j}^{2}$ -splitting mappings, for which there exists $x_{j}\in B_{1}(p_{j})$ and radii $r_{j}\geq s^{\delta_{j}}(x_{j})$ , such that there is no matrix $A$ such that $A\circ u:B_{r_{j}}(x_{j})\to\mathds{R}^{k}$ is an $\epsilon$ -splitting map. Without loss of generality, we can assume $r_{j}$ is the supremum of those radii for which there is no such matrix. In particular, there exists such a matrix $A_{j}$ corresponding to the radius $2r_{j}$ . Observe that $r_{j}\to 0$ . Indeed, we can see this just by using the identity map $A=I$ , since $\delta_{j}\to 0$ and $u:B_{2}(p)\to\mathds{R}^{2}$ is a $\delta_{j}^{2}$ -splitting map.

Now, consider the rescaled spaces $(M^{n}_{j},g^{\prime}_{j},x_{j})$ with $g^{\prime}_{j}\equiv r_{j}^{-2}g$ , and let $v_{j}\equiv A_{j}\circ\big(u_{j}-u_{j}(x_{j})\big):B_{2r^{-1}_{j}}(x_{j})\to\mathds{R}^{k}$ be a harmonic function on this space. We have normalized so that $v(x_{j})=0$ .

We have that $v_{j}:B_{2}(x_{j})\to\mathds{R}^{k}$ is an $\epsilon$ -splitting, and indeed for any $2\leq r\leq 2r_{j}^{-1}$ there exists some matrix $A_{r}$ such that $A_{r}\circ v:B_{r}(x_{j})\to\mathds{R}^{k}$ is an $\epsilon$ -splitting.

Note: Throughout the remainder of the argument, when there is no danger of confusion, for ease of notation, we will sometimes omit the subscript $j$ from various quantities including $v$ and $A$ , which in actuality depend on $j$ . For example, we omit the subcript $j$ from the matrices $A_{r},\,A_{2r}$ in Claim 1 below.

We will now break the proof into a series of claims.

Claim 1: For each $2\leq r\leq r_{j}^{-2}$ we have

(1-C(n)\epsilon)A_{2r}\leq A_{r}\leq(1+C(n)\epsilon)A_{2r}\,.

(3.38)

The defining properties of the matrices $A_{2r}$ is that they are lower triangular and that $A_{2r}\circ v:B_{2r}(x_{j})\to\mathds{R}^{k}$ is an $\epsilon$ -splitting map. In particular, we have the estimate

\displaystyle(2r)^{2}\fint_{B_{2r}(x_{j})}\big|\langle\nabla(A_{2r}\circ v)_{a},\nabla(A_{2r}\circ v)_{b}\rangle-\delta_{ab}\big|<\epsilon\,,

(3.39)

and thus, by doubling of the volume measure, we have

\displaystyle r^{2}\fint_{B_{r}(x_{j})}\big|\langle\nabla(A_{2r}\circ v)_{a},\nabla(A_{2r}\circ v)_{b}\rangle-\delta_{ab}\big|<C(n)\epsilon\,.

(3.40)

However in addition we also have

\displaystyle r^{2}\fint_{B_{r}(x_{j})}\big|\langle\nabla(A_{r}\circ v)_{a},\nabla(A_{r}\circ v)_{b}\rangle-\delta_{ab}\big|<\epsilon\,.

(3.41)

Combining this with the assumption that both $A_{r},\,A_{2r}$ are lower triangular proves the claim. $\square$

Now let us record some very important consequences of Claim 1. First, since by our normalization, $A_{2}\equiv I$ , we have for $r\geq 2$ the sublinear growth estimate

\displaystyle|A_{r}|,\,|A^{-1}_{r}|\leq r^{C(n)\epsilon}\,.

(3.42)

In particular, since $A_{r}\circ v:B_{r}(x_{j})\to\mathds{R}^{k}$ is an $\epsilon$ -splitting, and hence $\sup_{B_{r}(x_{j})}|\nabla(A_{r}\circ v)|\leq 1+\epsilon$ , we have for any $2\leq r\leq r_{j}^{-1}$ the sublinear growth conditions

		$\displaystyle\sup_{B_{r}(x_{j})}\|\nabla v^{a}_{j}\|\leq(1+C\epsilon)r^{C\epsilon}\,,$
		$\displaystyle\sup_{B_{r}(x_{j})}\|\omega_{j}\|\leq(1+C\epsilon)r^{C\epsilon}\,,$
		$\displaystyle r^{2}\fint_{B_{r}(x_{j})}\|\nabla^{2}v^{a}_{j}\|^{2}\leq C\epsilon r^{C\epsilon}\,,$		(3.43)

where $\omega_{j}\equiv dv_{j}^{1}\wedge\cdots\wedge dv_{j}^{k}$ is the pullback $k$ -form.

[01Y9]

Remark 3.1.

The sublinearity of the growth estimates in (3.43) will play a fundamental role in the proof; see in particular, Claims 3–5.

Our first application of these estimates is the following, which uses the induction statement to conclude that $v_{j}^{1},\ldots,v_{j}^{k-1}$ are improving in their splitting behavior as $j\to\infty$ .

Claim 2: There exists a lower triangular matrix $A$ such that $A\circ v:B_{2}(x_{j})\to\mathds{R}^{k}$ is a $C(n)\epsilon$ -splitting while for each $R>0$ the restricted map $A\circ v:B_{R}(x_{j})\to\mathds{R}^{k-1}$ , obtained by dropping the last function, is an $\epsilon_{j}(R)$ -splitting map, where $\epsilon_{j}(R)\to 0$ .

To prove the claim let us first denote by $\tilde{v}:B_{2r_{j}^{-1}}(x_{j})\to\mathds{R}^{k-1}$ the map obtained by dropping the last function $v^{k}$ . By our induction hypothesis there exists for every $r\geq 2$ an lower triangular matrix $\tilde{A}_{r}\in GL(k-1)$ such that $\tilde{A}_{r}\circ\tilde{v}:B_{r}(x_{j})\to\mathds{R}^{k-1}$ is an $\epsilon_{j}$ -splitting map with $\epsilon_{j}\to 0$ . Since both $\tilde{v}$ and $\tilde{A}_{2}\circ\tilde{v}$ are in particular $\epsilon$ -splittings on $B_{2}(x_{j})$ with $\tilde{A}_{2}$ lower triangular, then arguments similar to those in Claim 1 give $|\tilde{A}_{2}-I|<C(n)\epsilon$ , and the growth estimates

	$\displaystyle\sup_{B_{r}(x_{j})}\|\nabla(\tilde{A}_{2}\circ\tilde{v})\|$	$\displaystyle\leq(1+C\epsilon_{j})r^{C\epsilon_{j}}\,,$
	$\displaystyle r^{2}\fint_{B_{r}(x_{j})}\|\nabla^{2}(\tilde{A}_{2}\circ\tilde{v})\|^{2}$	$\displaystyle\leq C\epsilon_{j}r^{C\epsilon_{j}}\,.$		(3.44)

In particular, we can use the Hessian estimate and a Poincaré inequality to conclude

$\displaystyle\Big\|\fint_{B_{2}}\big\|\langle\nabla(\tilde{A}_{2}\circ\tilde{v})^{a},$	$\displaystyle\nabla(\tilde{A}_{2}\circ\tilde{v})^{b}\rangle-\delta^{ab}\big\|-\fint_{B_{R}}\big\|\langle\nabla(\tilde{A}_{2}\circ\tilde{v})^{a},\nabla(\tilde{A}_{2}\circ\tilde{v})^{b}\rangle-\delta^{ab}\big\|\Big\|$
	$\displaystyle\leq\fint_{B_{2}}\Big\|\|\langle\nabla(\tilde{A}_{2}\circ\tilde{v})^{a},\nabla(\tilde{A}_{\circ}2\tilde{v})^{b}\rangle-\delta^{ab}\big\|-\fint_{B_{R}}\big\|\langle\nabla(\tilde{A}_{2}\circ\tilde{v})^{a},\nabla(\tilde{A}_{2}\circ\tilde{v})^{b}\rangle-\delta^{ab}\big\|\Big\|$
	$\displaystyle\leq C(n,R)\fint_{B_{R}}\Big\|\|\langle\nabla(\tilde{A}_{2}\circ\tilde{v})^{a},\nabla(\tilde{A}_{2}\circ\tilde{v})^{b}\rangle-\delta^{ab}\big\|-\fint_{B_{R}}\big\|\langle\nabla(\tilde{A}_{2}\circ\tilde{v})^{a},\nabla(\tilde{A}_{2}\circ\tilde{v})^{b}\rangle-\delta^{ab}\big\|\Big\|$
	$\displaystyle\leq C(n,R)\fint_{B_{R}}\big\|\nabla\langle\nabla(\tilde{A}_{2}\circ\tilde{v})^{a},\nabla(\tilde{A}_{2}\circ\tilde{v})^{b}\rangle\big\|\leq\epsilon_{j}(R)\to 0\,.$	(3.45)

Thus for each $R>0$ we have $\tilde{A}_{2}\circ\tilde{v}:B_{R}(x_{j})\to\mathds{R}^{k-1}$ is an $\epsilon_{j}(R)$ -splitting. Finally, if we let $A=\tilde{A}_{2}\oplus 1$ act on $\mathds{R}^{k}$ by fixing the last component then we have proved the claim. $\square$

Note: We will from time to time in the proof replace $v$ by $A\circ v$ , where $A$ is a lower triangular matrix with $|A-I|<C(n)\epsilon$ . In particular, from this point on in the proof, we will assume $v^{a}$ has been normalized as in Claim 2. Thus $v^{a}:B_{2}(x_{j})\to\mathds{R}^{k}$ will be taken to be an $C\epsilon$ -splitting, while $v^{a}:B_{R}(x_{j})\to\mathds{R}^{k-1}$ is an $\epsilon_{j}(R)$ -splitting map.

A useful consequence is that we have for each $R>0$ and $1\leq\ell\leq k-1$ that

\displaystyle\fint_{B_{R}(x_{j})}|\nabla^{2}v^{a}|^{2}\leq\epsilon_{j}(R)\to 0\,.

(3.46)

[01YA]

Remark 3.2.

By way of orientation, we mention at this point that our long term goal is to show

\fint_{B_{R}(x_{j})}|\nabla^{2}v^{k}_{j}|^{2}\leq\epsilon_{j}(R)\to 0\,

which is the content of Claim 6. Once this has been achieved, the proof will be virtually complete.

Our next goal is to study in more detail the properties of $\omega_{j}=\omega^{k}_{j}=dv^{1}_{j}\wedge\cdots\wedge dv^{k}_{j}$ . First, since

\displaystyle\nabla\omega_{j}=\nabla(dv^{1}_{j})\wedge\cdots\wedge dv^{k}_{j}+\cdots+dv^{1}_{j}\wedge\cdots\wedge\nabla(dv^{k}_{j})\,,

(3.47)

we can use (3.43) to obtain for $2\leq r\leq r_{j}^{-1}$ that

\displaystyle r^{2}\fint_{B_{r}(x_{j})}|\nabla\omega_{j}|^{2}\leq C\epsilon\,r^{C\epsilon}\,.

(3.48)

Recall that our underlying assumptions are that we have for every $r\geq 1$ the estimate

\displaystyle r^{2}\fint_{B_{r}(x_{j})}|\Delta|\omega_{j}|\,|\leq\delta_{j}\fint_{B_{r}(x_{j})}|\omega_{j}|\,.

(3.49)

By combining this with (3.43), we get that for every $2\leq r\leq r_{j}^{-1}$ ,

\displaystyle r^{2}\fint_{B_{r}(x_{j})}|\Delta|\omega_{j}|\,|\leq C\delta_{j}r^{C\epsilon}\,.

(3.50)

Now we are ready to make our third claim:

Claim 3: For each fixed $R\geq 1$ , we have $\fint_{B_{R}(x_{j})}\big||\omega_{j}|^{2}-\fint_{B_{R}(x_{j})}|\omega_{j}|^{2}\big|\to 0$ .

The proof of Claim 3 will rely on the sublinear growth estimates (3.43), (3.48), (3.50), standard heat kernel estimates for almost nonnegative Ricci curvature, (3.55)–(3.57) and the Bakry-Emery gradient estimate for the heat kernel (3.64). In particular, the sublinear growth condition in (3.48) enters crucially in (3.63) and its consequence (3.66).

Fix $R\geq 1$ and consider the maximal function

\displaystyle M^{R}(x)\equiv\sup_{r\leq R}\fint_{B_{r}(x)}|\Delta|\omega_{j}||\,,

(3.51)

for $x\in B_{R}(x_{j})$ . Since by the Bishop-Gromov inequality, the Riemannian measure is doubling, we can combine the usual maximal function arguments with (3.50) and conclude that there exists a subset $U_{j}\subseteq B_{R}(x_{j})$ such that

		$\displaystyle\frac{{\rm Vol}(B_{R}(x_{j})\setminus U_{j})}{{\rm Vol}(B_{R}(x_{j}))}\leq\epsilon_{j}(R)\to 0\,,$
		$\displaystyle M^{R}(x)\leq\epsilon_{j}(R)\to 0\,,$		(3.52)

for all $x\in U_{j}$ . Relation (3.52) will be used in (3.59).

As a point of notation, we mention that below, the symbol, $\epsilon_{j}(R)$ , will always denote a quantity, regardless of origin, satisfying $\epsilon_{j}(R)\to 0$ when $j\to\infty$ with $R$ fixed. Likewise for the symbol $\epsilon_{j}(S)$ .

Now let $\varphi\geq 0$ be a smooth cutoff function as in [ChCo1], such that $\varphi\equiv 1$ on $B_{r_{j}^{-1}/2}(p)$ , ${\rm supp}(\varphi)\subset B_{r_{j}^{-1}}(p)$ , and such that $r^{-1}_{j}|\nabla\varphi|,r_{j}^{-2}|$ $\Delta\varphi|\leq C(n)$ . For $x\in B_{R}(x_{j})$ let us consider the function

\displaystyle\int|\omega_{j}|\varphi\rho_{t}(x,dy)\,,

(3.53)

where $\rho_{t}$ is the heat kernel centered at $x$ . Then we have the equality

\displaystyle\frac{d}{dt}\int|\omega_{j}|\varphi\rho_{t}(x,dy)=\int\Big(\Delta|\omega_{j}|\varphi+\langle\nabla|\omega_{j}|,\nabla\varphi\rangle+|\omega_{j}|\Delta\varphi\Big)\rho_{t}(x,dy)\,.

(3.54)

As a consequence of our assumption that ${\rm Ric}_{M^{n}_{j}}\geq-\delta^{2}_{j}r_{j}^{2}$ , we have the usual heat kernel estimates [SY]

\rho_{t}(x,y)\leq C(n){\rm Vol}(B_{\sqrt{t}}(x))^{-1/2}{\rm Vol}(B_{\sqrt{t}}(y))^{-1/2}e^{-\frac{d^{2}(x,y)}{2t}+C(n)\delta^{2}_{j}r_{j}^{2}t}\,,

(3.55)

which implies that for $y\in B_{r_{j}^{-1}}(x)$ and $t\leq r_{j}^{-2}$ , we have

\rho_{t}(x,y)\leq C(n){\rm Vol}(B_{\sqrt{t}}(x))^{-1/2}{\rm Vol}(B_{\sqrt{t}}(y))^{-1/2}e^{-\frac{d^{2}(x,y)}{2t}}\,.

(3.56)

We can use the volume doubling and monotonicity property to observe the following useful inequality. If $y\in B_{r}(x)$ , then

	$\displaystyle\rho_{t}(x,y)$	$\displaystyle\leq C(n)\Big(\frac{{\rm Vol}(B_{r}(x))}{{\rm Vol}(B_{\sqrt{t}}(x))^{1/2}{\rm Vol}(B_{\sqrt{t}}(y))^{1/2}}\Big){\rm Vol}(B_{r}(x))^{-1}e^{-\frac{d^{2}(x,y)}{2t}}$
		$\displaystyle\leq C(n)\Big(\frac{r}{t^{1/2}}\Big)^{n}{\rm Vol}(B_{r}(x))^{-1}e^{-\frac{d^{2}(x,y)}{2t}}\,.$		(3.57)

Let us fix $S>>R\geq 2$ and consider times $0<t\leq S^{2}$ . By combining the heat kernel estimate, (3.57), with the growth estimates (3.43), (3.48), for all $x\in B_{R}(x_{j})$ and $0<t\leq S^{2}$ , we can bound the second two terms of the last equation by

$\displaystyle\int\big\|\langle\nabla\|\omega_{j}\|,\nabla\varphi\rangle+\|\omega_{j}\|\Delta\varphi\|\rho_{t}(x,dy)$	$\displaystyle=\int_{A_{r_{j}^{-1}/2,r_{j}^{-1}}(x_{j})}\big\|\langle\nabla\|\omega_{j}\|,\nabla\varphi\rangle+\|\omega_{j}\|\,\|\Delta\varphi\|\big\|\rho_{t}(x,dy)\,$
	$\displaystyle\leq Cr_{j}r_{j}^{C\epsilon}{\rm Vol}(B_{r_{j}}(x_{j})){\rm Vol}(B_{\sqrt{t}}(x))^{-1/2}{\rm Vol}(B_{\sqrt{t}}(y))^{-1/2}\,e^{-\frac{1}{2t}r^{-2}_{j}}$
	$\displaystyle\leq Cr_{j}^{2+C\epsilon}\Big(\frac{r_{j}}{t^{1/2}}\Big)^{n}e^{-\frac{1}{2t}r^{-2}_{j}}\leq\epsilon_{j}(S)\to 0\,.$	(3.58)

Note that the sublinear growth in (3.43), (3.48), is not crucial here. Polynomial growth would suffice.

To estimate the first term of (3.54) is more involved. To this end, we begin with an estimate in which we must restrict attention to points $x\in U_{j}\subseteq B_{R}(x_{j})$ ; see (3.52). Below, we write $t=r^{2}$ and so, we consider $0<r<S$ . We also put $r^{\alpha}=2^{\alpha}r$ . Suppose first that $\sqrt{t}=r\leq R$ . Then we have

$\displaystyle\int\big\|\Delta\|\omega_{j}\|\big\|\varphi\rho_{r^{2}}(x,dy)$	$\displaystyle=\int_{B_{r}(x)}\big\|\Delta\|\omega_{j}\|\big\|\varphi\rho_{r^{2}}(x,dy)+\sum_{\alpha}\int_{A_{r^{\alpha},r^{\alpha+1}}(x)}\big\|\Delta\|\omega_{j}\|\big\|\varphi\rho_{r^{2}}(x,dy)\,$	(3.59)
	$\displaystyle\leq C(n)\fint_{B_{r}(x)}\big\|\Delta\|\omega_{j}\|\,\big\|+C(n)\sum_{\alpha}\Big(\frac{r_{\alpha}}{r}\Big)^{n}e^{-\big(r^{-1}r^{\alpha}\big)^{2}}\fint_{B_{2^{\alpha}r}(x)}\big\|\Delta\|\omega_{j}\|\,\big\|$
	$\displaystyle\leq C(n)\fint_{B_{r}(x)}\big\|\Delta\|\omega_{j}\|\,\big\|+C(n)\sum_{\alpha}2^{n\alpha}e^{-2^{2\alpha}}\fint_{B_{2^{\alpha}r}(x)}\big\|\Delta\|\omega_{j}\|\,\big\|$
	$\displaystyle=C(n)\fint_{B_{r}(x)}\big\|\Delta\|\omega_{j}\|\big\|+C(n)\sum_{r^{\alpha}\leq R}2^{n\alpha}e^{-2^{2\alpha}}\fint_{B_{2^{\alpha}r}(x)}\big\|\Delta\|\omega_{j}\|\,\big\|+C(n)\sum_{r^{\alpha}>R}2^{n\alpha}e^{-2^{2\alpha}}\fint_{B_{2^{\alpha}r}(x)}\big\|\Delta\|\omega_{j}\|\,\big\|\,$
	$\displaystyle\leq C\epsilon_{j}(R)+C\sum_{r^{\alpha}\leq R}2^{n\alpha}e^{-2^{2\alpha}}\epsilon_{j}(R)+CR^{-2}\sum_{r^{\alpha}>R}2^{n\alpha}e^{-2^{2\alpha}}\delta_{j}\to 0\,.$

Note that in estimating the first two terms in the last line of (3.59) we use the maximal function estimate (3.52), which is the reason for restricting attention to $x\in U_{j}$ . For the third term in the last line we use (3.50).

Similarly, $\sqrt{t}=r>R$ , the first two terms on the last line of (3.59) are absent and we just get

\displaystyle\int\big|\Delta|\omega_{j}|\big|\varphi\rho_{r^{2}}(x,dy)\leq CR^{-2}\sum_{\alpha}2^{n\alpha}e^{-2^{2\alpha}}\delta_{j}\to 0\,.\

(3.60)

By combining (3.54), (3.58), (3.59), (3.60), we get for $x\in U_{j}$ and $0<t\leq S^{2}$ ,

\displaystyle\Big|\frac{d}{dt}\int|\omega_{j}|\varphi\rho_{t}(x,dy)\Big|\leq\epsilon_{j}(S)\to 0\,,

(3.61)

uniformly in $U_{j}$ .

At this point, by using (3.61) and integrating with respect to $t$ from $0$ to $S^{2}$ , we have for any $x\in U_{j}\subseteq B_{R}(x_{j})$ ,

\displaystyle\Big||\omega_{j}|(x)-\int|\omega_{j}|\varphi\rho_{S^{2}}(x,dy)\Big|\leq\epsilon_{j}(S)\cdot S^{2}\to 0\,,

(3.62)

uniformly in $U_{j}$ .

By arguing in a manner similar to the above (but without the need for a maximal function estimate) we can use (3.48), to see that for all $x\in B_{2R}(x_{j})$

$\displaystyle\int\big\|\nabla(\|\omega_{j}\|\varphi)\big\|^{2}\rho_{S^{2}}(x,dy)$	$\displaystyle\leq 2\int\|\nabla\omega_{j}\|^{2}+\|\omega_{j}\|^{2}\|\nabla\varphi\|^{2}\rho_{S^{2}}(x,dy)$
	$\displaystyle\leq C\sum 2^{n\alpha}e^{-2^{2\alpha}}\fint_{B_{2^{\alpha}S}(x)}\|\nabla\omega_{j}\|^{2}+Cr_{j}^{2-C\epsilon}\Big(\frac{r_{j}}{S}\Big)^{n}e^{-\frac{1}{S^{2}}r_{j}^{-2}}\,$
	$\displaystyle\leq C\,S^{-2+C\epsilon}+\epsilon_{j}(S)\,,$	(3.63)

where without loss of generality, we can assume that our original $\epsilon$ has been chosen so that $-2+C\epsilon<0$ . As previously mentioned, it is at just this point that the sublinearity in (3.48) has entered crucially, giving rise to the negative power of $S$ in (3.63), which comes to fruition in (3.66).

We have that $H_{t}\big(|\omega_{j}|\varphi\big)=\int|\omega_{j}|\varphi\rho_{t}(x,dy)$ solves the heat equation. So using the Bakry-Emery gradient estimate, [BE85], we have for any $x\in B_{2R}(x_{j})$

\displaystyle|\nabla H_{t}\big(|\omega_{j}|\varphi\big)|^{2}(x)\leq e^{\delta_{j}^{2}r_{j}^{2}t}H_{t}|\nabla(|\omega_{j}|\varphi)|^{2}(x)\,.

(3.64)

In particular, using (3.63) we have

\displaystyle{}\sup_{B_{2R}(x_{j})}\Big|\nabla_{x}\int|\omega_{j}|\varphi\rho_{S^{2}}(x,dy)\Big|\leq\frac{C}{S^{1-C\epsilon/2}}+\epsilon_{j}(S)\,.

(3.65)

Combining this with (3.62) we get for any pair of points, $x,y\in U_{j}$ ,

$\displaystyle\big\|\,\|\omega_{j}\|(x)-\|\omega_{j}\|(y)\big\|$	$\displaystyle\leq\big\|\omega_{j}(x)-\int\|\omega_{j}\|\varphi\rho_{S^{2}}(x,dz)\big\|+\big\|\omega_{j}(y)-\int\|\omega_{j}\|\varphi\rho_{S^{2}}(y,dz)\big\|$
	$\displaystyle+\big\|\int\|\omega_{j}\|\varphi\rho_{S^{2}}(x,dy)-\int\|\omega_{j}\|\varphi\rho_{S^{2}}(y,dz)\big\|$
	$\displaystyle\leq\epsilon_{j}(S)+\frac{CR}{S^{1-C\epsilon/2}}\,.$	(3.66)

By letting $S$ tend to infinity sufficiently slowly, we get for $x,y\in U_{j}$ , that

\displaystyle\big|\,|\omega_{j}|(x)-|\omega_{j}|(y)\big|

\displaystyle\leq\epsilon_{j}(R)\to 0\,.

(3.67)

Finally, to finish the proof, we use the supremum bound (3.48) on $|\omega|$ to note that for $x\in U_{j}$ , we have

$\displaystyle\Big\|\fint_{B_{R}(x_{j})}\|\omega_{j}\|^{2}-\|\omega_{j}\|^{2}(x)\Big\|$	$\displaystyle\leq\fint_{B_{R}(x_{j})}\Big\|\|\omega_{j}\|^{2}-\|\omega_{j}\|^{2}(x)\Big\|$
	$\displaystyle\leq\fint_{B_{R}(x_{j})}\big\|\|\omega_{j}\|-\|\omega_{j}\|(x)\big\|\cdot\big\|\|\omega_{j}\|+\|\omega_{j}\|(x)\big\|$	(3.68)
	$\displaystyle\leq C(n,R)\fint_{B_{R}(x_{j})}\big\|\|\omega_{j}\|-\|\omega_{j}\|(x)\big\|$
	$\displaystyle\leq C(n,R)\fint_{U_{j}}\big\|\|\omega_{j}\|-\|\omega_{j}\|(x)\big\|+C(n,R)\fint_{B_{R}(x_{j})\setminus U_{j}}\big\|\|\omega_{j}\|-\|\omega_{j}\|(x)\big\|$
	$\displaystyle\leq C(n,R)\epsilon_{j}(R)+C(n,R)\cdot\frac{{\rm Vol}(B_{R}(x_{j})\setminus U_{j})}{{\rm Vol}(B_{R}(x_{j}))}\to 0\,.$	(3.69)

Hence, we have

\displaystyle\fint_{B_{R}(x_{j})}\Big|\,|\omega_{j}|^{2}-\fint_{B_{R}(x_{j})}|\omega_{j}|^{2}\Big|\leq\Big|\fint_{B_{R}(x_{j})}|\omega_{j}|^{2}-|\omega_{j}|^{2}(x)\Big|+\fint_{B_{R}(x_{j})}\Big||\omega_{j}|^{2}-|\omega_{j}|^{2}(x)\Big|\to 0\,,

(3.70)

which proves the claim. $\square$

We know from (3.43), (3.48) that $|\nabla\omega_{j}|$ has $L^{2}$ bounds. It is crucial to improve these to bounds that are small compared to $\epsilon$ . This is the content of the next claim:

Claim 4: For fixed $R$ we have

\fint_{B_{R}(x_{j})}|\nabla\omega_{j}|^{2}\leq\epsilon_{j}(R)\to 0\,.

(3.71)

To see this fix $R$ and as in [ChCo1], let $\varphi:B_{2R}(x_{j})\to\mathds{R}^{+}$ be a cutoff function with $\varphi\equiv 1$ on $B_{R}(x_{j})$ and $R|\nabla\varphi|,\,R^{2}|\Delta\varphi|\leq C(n)$ . We use the Bochner formula

$\displaystyle\Delta\|\omega_{j}\|^{2}$	$\displaystyle=2\|\nabla\omega_{j}\|^{2}+2\langle\sum_{b}dv^{1}_{j}\wedge\cdots{\rm Ric}(dv^{b}_{j})\wedge\cdots\wedge dv^{k}_{j},\omega\rangle$
	$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\langle\sum_{a\neq b}dv^{1}_{j}\wedge\nabla^{c}(dv^{a}_{j})\wedge\cdots\wedge\nabla_{c}(dv^{b}_{j})\wedge\cdots\wedge dv^{k}_{j},\omega_{j}\rangle$
	$\displaystyle\geq 2\|\nabla\omega_{j}\|^{2}-C(n)\delta_{j}^{2}r_{j}^{2}\|\omega_{j}\|^{2}-C(n)\|\nabla(dv)\|^{2}\|\omega_{j}\|^{2}$
	$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\langle\sum_{a\neq b}dv^{1}_{j}\wedge\nabla^{c}(dv^{a}_{j})\wedge\cdots\wedge\nabla_{c}(dv^{b}_{j})\wedge\cdots\wedge dv^{k}_{j},\omega_{j}\rangle\,,$	(3.72)

which together with the growth estimates (3.43) allows us to compute

$\displaystyle\fint_{B_{R}(x_{j})}\|\nabla\omega_{j}\|^{2}$	$\displaystyle\leq C(n)\fint_{B_{2R}(x_{j})}\varphi\Delta\|\omega_{j}\|^{2}+C(n,R)\sum_{a\neq b}\fint_{B_{2R}(x_{j})}\|\nabla^{2}v^{a}_{j}\|\,\|\nabla^{2}v^{b}\|+C(n,R)\delta_{j}^{2}r_{j}^{2}$
	$\displaystyle\leq C\fint_{B_{2R}(x_{j})}\Delta\varphi\,\big(\|\omega_{j}\|^{2}-\fint_{B_{2R}(x_{j})}\|\omega_{j}\|^{2}\big)$
	$\displaystyle\,\,\,\,\,\,+C(n,R)\sum_{a\neq b}\Big(\fint_{B_{2R}(x_{j})}\|\nabla^{2}v^{a}\|^{2}\Big)^{1/2}\Big(\fint_{B_{2R}(x_{j})}\|\nabla^{2}v^{b}\|^{2}\Big)^{1/2}+\epsilon_{j}(R)$
	$\displaystyle\leq C\fint_{B_{2R}(x_{j})}\big\|\|\omega_{j}\|^{2}-\fint_{B_{2R}(x)}\|\omega_{j}\|^{2}\big\|+\epsilon_{j}(R)\leq\epsilon_{j}(R)\to 0\,,$	(3.73)

where we have used Claim 3 and (3.46). Note that it is important that we have $a\neq b$ in the summation, so that at least one of the Hessian terms in each factor is going to zero as $j\to\infty$ . This proves the claim.∎

As mentioned in Remark 3.2, to complete the proof we must show that $\fint_{B_{R}(x_{j})}|\nabla^{2}v^{k}_{j}|^{2}\to 0$ as $j\to\infty$ . To prove this we will first pass to limits and obtain information on the limiting space. That is, we have been considering a sequence $(M^{n}_{j},d_{j},x_{j})$ with ${\rm Ric}_{M^{n}_{j}}\geq-\delta_{j}^{2}r_{j}^{2}\to 0$ . After passing to a subsequence if necessary, we can take a measured pointed Gromov-Hausdorff limit

\displaystyle(M^{n}_{j},d^{\prime}_{j},x_{j})\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}(X,d,x)\,,

(3.74)

to obtain an $RCD(n,0)$ space $X$ , see [AGS12], [AGS12-2]. The fact that $X$ is an $RCD(n,0)$ space is used below in applying the mean value estimate (3.84), which is known to hold for such spaces.

In addition, we can assume that the functions $v_{j}^{\ell}$ converge to harmonic functions.

\displaystyle v_{j}^{\ell}\to v^{\ell}:X\to\mathds{R}\,.

(3.75)

Indeed, for any ball $B_{R}(x_{j})$ we can characterize $v^{\ell}_{j}$ as minimizers of the Dirichlet energy with fixed Dirichlet boundary values. Our assertion then follows from the lower semicontinuity of the Dirichlet energy [AGS12-2] combined with the Mosco convergence of the Dirichlet form [GMS14], to see that the limit also minimizes the Dirichlet energy on any ball.

Observe first, that by using Claim 2 and Lemma 1.7, we have

\displaystyle X=\mathds{R}^{k-1}\times Y\,,

(3.76)

where $v^{1},\ldots,v^{k-1}:X\to\mathds{R}$ are linear functions which induce the $\mathds{R}^{k-1}$ factor and we can identify $Y=(v^{1},\ldots,v^{k-1})^{-1}(0^{k-1})$ . We are left with understanding the behavior of $v^{k}$ . We will seein Claim 6 that it too is linear, and in the process prove our Hessian estimate. We first show the following:

Claim 5: There exists $a_{1},\ldots,a_{k-1}\in\mathds{R}$ with $|a_{\ell}|<C(n)\epsilon$ such that $v^{k}-a_{1}v^{1}-\cdots-a_{k-1}v^{k-1}:X\to\mathds{R}$ is a function of only the $Y$ variable.

To prove the claim let us fix any vector $V\in\mathds{R}^{k-1}$ and consider the map $Dv^{k}:X\to\mathds{R}$ defined by

\displaystyle Dv^{k}(y)=v^{k}(y+V)-v^{k}(y)\,,

(3.77)

where of course, the translation $x\to x+V$ is well defined, since $X\equiv\mathds{R}^{k-1}\times Y$ . The function $v^{k}(y)$ is harmonic, and the translation map $x\to x+V$ is a measure preserving isometry. Thus, $v^{k}(x+V)$ is a harmonic function as well. Since $X$ is an $RCD$ space, and hence the Laplacian $\Delta$ on $X$ is linear, it follows that $Dv^{k}$ is harmonic. Using the estimates (3.43) we have the growth condition

\displaystyle\sup_{B_{r}(x)}|Dv^{k}|\leq C|V|^{1+C\epsilon}\cdot r^{C\epsilon}\,.

(3.78)

This is to say that $Dv^{k}$ is a harmonic function with sublinear growth. It follows that $Dv^{k}$ must be a constant. Indeed, let $\varphi$ be a cutoff on $B_{2R}$ with $\varphi\equiv 1$ on $B_{R}(x)$ and $|\nabla\varphi|\leq 10R^{-1}$ . Then on the one hand, we have since $Dv^{k}$ is harmonic and the Dirichlet form is bilinear that

	$\displaystyle 0$	$\displaystyle=\fint_{B_{2R}(x)}\langle\nabla Dv^{k},\nabla(\varphi^{2}Dv^{k})\rangle$		(3.79)
		$\displaystyle=\fint_{B_{2R}(x)}\varphi^{2}\|\nabla Dv^{k}\|^{2}+2\fint_{B_{2R}(x)}\varphi\,Dv^{k}\,\langle\nabla Dv^{k},\nabla\varphi\rangle\,.$		(3.80)

By rearranging terms, we obtain

$\displaystyle\fint_{B_{R}(x)}\|\nabla Dv^{k}\|^{2}$	$\displaystyle\leq\fint_{B_{2R}(x)}\varphi^{2}\|\nabla Dv^{k}\|^{2}$	(3.81)
	$\displaystyle\leq\frac{1}{2}\fint_{B_{2R}(x)}\varphi^{2}\|\nabla Dv^{k}\|^{2}+8\fint_{B_{2R}(x)}\|Dv^{k}\|^{2}\|\nabla\varphi\|^{2}$	(3.82)
	$\displaystyle\leq CR^{-2+C\epsilon}\,.$	(3.83)

On the other hand, ${\rm Ric}_{M^{n}_{j}}\geq-(n-1)\delta^{2}_{j}r_{j}^{2}\to 0$ and so $X$ is an $RCD(n,0)$ space. On such spaces, there is a mean value inequalilty for the norm squared of the gradient of a harmonic function; see for instance [MN14]. When applied to the harmonic function $Dv^{k}$ it gives for $r>0$ fixed and $R\to\infty$

\displaystyle\sup_{B_{r}(x)}|\nabla Dv^{k}|^{2}\leq C\fint_{B_{R}(x)}|\nabla Dv^{k}|^{2}\leq CR^{-2+C\epsilon}\to 0\,.

(3.84)

Note that once again we have exploited the sublinearity of the growth estimates. In particular, it now follows that $Dv^{k}$ is a constant. Since this holds for any $V\in\mathds{R}^{k-1}$ , we have that $v^{k}$ is linear in the $\mathds{R}^{k-1}$ variable. More precisely, since the $\mathds{R}^{k-1}$ factor is spanned by $v^{1},\ldots,v^{k-1}$ we have

\displaystyle v^{k}=v^{k}_{Y}+a_{1}v^{1}+\cdots+a_{k-1}v^{k-1}\,,

(3.85)

where $v^{k}_{Y}:Y\to\mathds{R}$ . Since $v_{j}\to v:X\to\mathds{R}^{k}$ are $C\epsilon$ -splittings on $B_{2}(x_{j})$ , we automatically have the bounds $|a_{\ell}|\leq C(n)\epsilon$ . This finishes the claim. $\square$

To complete the proof, we want to see that the Hessians of $v^{k}_{j}$ are tending to zero as $j\to\infty$ . This is the content of Claim 6 below. However, prior to stating this claim, we will make some additional normalizations.

To begin with, we can use Claim 5 to further normalize the mappings $v_{j}$ by composing with another lower triangular matrix. Indeed, as a corollary of Claim 5 we may choose a lower triangular matrix $A$ with $|A-I|<C(n)\epsilon$ , and whose restriction to the first $(k-1)\times(k-1)$ terms is the identity, such that $Av_{j}:B_{2}(x_{j})\to\mathds{R}^{k}$ is still an $C(n)\epsilon$ -splitting, while $A\circ v^{k}_{j}\to A\circ v^{k}:\mathds{R}^{k-1}\times Y\to\mathds{R}$ is independent of the $\mathds{R}^{k-1}$ factor. Further, let us consider the induced form $A\circ\omega_{j}=d(A\circ v_{j}^{1})\wedge\cdots\wedge d(A\circ v^{k}_{j})=dv_{j}^{1}\wedge\cdots\wedge d(A\circ v^{k}_{j})$ . Then after multiplying the $k^{th}$ row of $A$ by a constant $c$ with $|c-1|\leq C(n)\epsilon$ we may further assume that

\displaystyle\fint_{B_{2}(x_{j})}|A\circ\omega_{j}|^{2}=1\,.

(3.86)

From this point forward in the proof, for ease of notation, we will write $v_{j}$ for what was denoted above by $A\circ v_{j}$ In particular, this $v_{j}$ differs from the original mapping $u_{j}$ only by composition with a lower triangular matrix. We will eventually see that $v_{j}:B_{1}(x_{j})\to\mathds{R}^{k}$ is an $\epsilon_{j}$ -splitting, which will give the desired contradiction and finish the proof.

Claim 6. For each $R>0$ , we have $\fint_{B_{R}(x_{j})}|\nabla^{2}v^{k}_{j}|^{2}\leq\epsilon_{j}(R)\to 0$ .

The fact that $v_{j}:B_{R}(x_{j})\to\mathds{R}^{k-1}$ is an $\epsilon_{j}(R)$ -splitting,

\fint_{B_{2}(x_{j})}|\omega_{j}|^{2}=1\,,

together with

\fint_{B_{R}(x_{j})}|\nabla\omega_{j}|^{2}\leq\epsilon_{j}(R)\to 0\,,

implies

\displaystyle\fint_{B_{R}(x_{j})}\big||\omega^{\ell}_{j}|-1\big|\leq\epsilon_{j}(R)\qquad({\rm for\,\,all}\,\,1\leq\ell\leq k)\,.

(3.88)

Now we will show that

\displaystyle\fint_{B_{R}(x_{j})}\big||\nabla v^{k}_{j}|^{2}-1\big|\leq\epsilon_{j}(R)\to 0\,.

(3.89)

Once this is accomplished, as we have done repeatedly, we can argue with Bochner’s formula to obtain the Hessian estimate in the claim.

Define the $1$ -form

\displaystyle V_{j}\equiv\langle\omega^{k-1}_{j},\omega_{j}\rangle\,.

(3.90)

Note $\omega^{k-1}_{j}\wedge V_{j}$ is proportional to $\omega_{j}=\omega^{k-1}_{j}\wedge dv^{k}_{j}=\omega^{k-1}_{j}\wedge\big(dv^{k}_{j}-\pi_{k-1}dv^{k}_{j}\big)$ . More generally, we have that $V_{j}\in\text{span}\{\nabla v^{1}_{j},\ldots,\nabla v^{k}_{j}\}$ is perpendicular to $\text{span}\{\nabla v^{1}_{j},\ldots,\nabla v^{k-1}_{j}\}$ . From the above, we get

\displaystyle\fint_{B_{R}(x_{j})}|V_{j}-\big(dv^{k}_{j}-\pi_{k-1}dv^{k}_{j}\big)|\leq\epsilon_{j}(R)\,.

(3.91)

On the other hand, by (3.73) we have

\displaystyle\fint_{B_{R}(x_{j})}|\nabla V_{j}|^{2}\leq\epsilon_{j}(R)\to 0\,,

(3.92)

and thus using (3.88) we have

\displaystyle\fint_{B_{R}(x_{j})}\big||V_{j}|-1\big|\leq\epsilon_{j}(R)\,.

(3.93)

Therefore, from (3.91) we get

\displaystyle\fint_{B_{R}(x_{j})}\Big||dv^{k}_{j}-\pi_{k-1}dv^{k}_{j}|-1\Big|\leq\epsilon_{j}(R)\to 0\,.

(3.94)

It follows that our main concern is to show $|\pi_{k-1}(dv^{k}_{j})|\to 0$ as $j\to\infty$ .

For $R>0$ , we can use the segment inequality of [ChCo1] along with (3.54) to find a subset $U_{R}\subseteq B_{R}(x_{j})$ with

\displaystyle{\rm Vol}(B_{R}(x_{j})\setminus U_{R})\leq\epsilon_{j}(R)\to 0\,,

(3.95)

such that for each $x\in U_{R}$ there exists a subset $U_{R}(x)\subseteq B_{2R}(x_{j})$ with

\displaystyle{\rm Vol}(B_{R}(x_{j})\setminus U_{R}(x))\leq\epsilon_{j}(R)\to 0\,,

(3.96)

and such that if $y\in U_{R}(x)$ then there is a unique geodesic $\gamma$ connecting $x$ and $y$ , and for $1\leq\ell\leq k-1$ , we have the estimates

		$\displaystyle\int_{\gamma}\|\nabla^{2}v^{\ell}_{j}\|^{2}\leq\epsilon_{j}(R)\,,$
		$\displaystyle\int_{\gamma}\|\nabla^{2}v^{k}_{j}\|^{2}\leq C\epsilon\,.$		(3.97)

Now, for $x\in U_{R}$ let us choose $x_{1},\ldots,x_{k-1}\in U_{R}(x)$ such that

		$\displaystyle\|v^{\ell}_{j}(x)-v^{\ell}_{j}((x_{\ell})-1\|<\epsilon_{j}(R)\,,$
		$\displaystyle\|v^{k}_{j}((x)-v^{k}_{j}((x_{\ell})\|<\epsilon_{j}(R)\,,$
		$\displaystyle\big\|\|\nabla v^{\ell}_{j}\|-1\big\|(x_{\ell})\leq\epsilon_{j}(R)\,.$		(3.98)

Note that the geodesic $\gamma_{\ell}$ connecting $x$ and $x_{\ell}$ is contained in the approximate $\mathds{R}^{k-1}$ factor from the splitting induced by $v^{1}_{j},\ldots,v^{k-1}_{j}$ . Thus, we get the pointwise estimate

\displaystyle|v^{k}_{j}((x)-v^{k}_{j}(\gamma_{\ell}(t))|<\epsilon_{j}(R)\,,

(3.99)

along all of $\gamma_{\ell}$ . In particular, for every interval $I\subseteq\gamma_{\ell}$ , we have

\displaystyle\big|\int_{I}\langle\nabla v^{k}_{j},\dot{\gamma}_{\ell}\rangle\big|<\epsilon_{j}(R)\,.

(3.100)

Thus, it follows from the mean value theorem that for each interval $I$ , there is a point $t_{I}\in I$ such that

\displaystyle|\langle\nabla v^{k}_{j}(\gamma(t_{I})),\dot{\gamma}(t_{I})\rangle|<\frac{\epsilon_{j}(R)}{|I|}\,.

(3.101)

Now let us use (3.97) and the Schwarz inequality to get

\displaystyle\int_{\gamma_{\ell}}\big|\nabla_{\dot{\gamma}_{\ell}}\langle\nabla v^{k}_{j},\dot{\gamma}_{\ell}\rangle\big|<C\epsilon^{\frac{1}{2}}|I|^{\frac{1}{2}}\,.

(3.102)

By combining this with (3.101), we get that for each interval $I$ ,

\displaystyle\sup_{I}\,|\langle\nabla v^{k}_{j},\dot{\gamma}\rangle|\leq C\cdot\frac{\epsilon_{j}(R)}{|I|^{\frac{1}{2}}}\,.

(3.103)

By covering $\gamma$ with intervals whose size decreases to zero sufficiently slowly as $j\to\infty$ , we get the pointwise estimate

\displaystyle{\sup_{I}}\,|\langle\nabla v^{k}_{j},\dot{\gamma}\rangle|\leq\epsilon_{j}(R)\to 0\,,

(3.104)

and in particular, at $x=\gamma(0)$ , we have

|\langle\nabla v^{k}_{j},\dot{\gamma}_{\ell}\rangle|(x)<\epsilon_{j}(R)\to 0\,.

For $1\leq\ell\leq k-1$ , we can argue similarly with $v^{\ell}_{j}$ in place of $v^{k}_{j}$ . Namely, by (3.97) and (3.98), we have

		$\displaystyle\big\|\int_{\gamma_{\ell}}\big(1-\langle\nabla v^{\ell}_{j},\dot{\gamma}_{\ell}\rangle\big)\,\big\|<\epsilon_{j}(R)\,,$
		$\displaystyle\int_{\gamma_{\ell}}\|\nabla_{\dot{\gamma}_{\ell}}\langle\nabla v^{\ell}_{j},\dot{\gamma}_{\ell}\rangle\|<\epsilon_{j}(R)\,.$		(3.105)

Thus, we get

\displaystyle\sup_{\gamma_{\ell}}|1-\langle\nabla v^{\ell}_{j},\dot{\gamma}_{\ell}\rangle|(\gamma_{\ell})<\epsilon_{j}(R)\,.

(3.106)

At $x=\gamma_{\ell}(0)$ , this leads to

\displaystyle|\dot{\gamma}_{\ell}-\pi_{k-1}\dot{\gamma}_{\ell}|(x)<\epsilon_{j}(R)\,.

(3.107)

From this together with (3.104) we get

\displaystyle|\pi_{k-1}(\nabla v^{k}_{j})|(x)<\epsilon_{j}(R)\,.

(3.108)

Since $|\nabla v^{k}_{j}|\leq C(R)$ , we obtain from (3.95) that

	$\displaystyle\fint_{B_{R}(x_{j})}\|\pi_{k-1}(\nabla v^{k}_{j})\|$	$\displaystyle={\rm Vol}(B_{R}(x_{j}))^{-1}\int_{U_{R}}\|\pi_{k-1}(\nabla v^{k}_{j})\|+{\rm Vol}(B_{R}(x_{j}))^{-1}\int_{B_{R}\setminus U_{R}}\|\pi_{k-1}(\nabla v^{k}_{j})\|$
		$\displaystyle\leq\epsilon_{j}(R)\to 0\,.$		(3.109)

By combining this with (3.94), we get the desired estimate

\displaystyle\fint_{B_{R}(x_{j})}\big||\nabla v^{k}_{j}|^{2}-1\big|\leq\epsilon_{j}(R)\to 0\,.

(3.110)

Since $v^{k}_{j}$ is harmonic we can now argue with Bochner’s formula as in the proof of (3.5), to obtain the Hessian estimate, $\fint_{B_{R}(x_{j})}|\nabla^{2}v^{k}_{j}|^{2}\leq\epsilon_{j}(R)\to 0$ . This completes the proof of the claim. $\square$

Now we can finish the proof of the Transformation theorem. Indeed, we will see that $v_{j}=A\circ u:B_{1}(x_{j})\to\mathds{R}^{k}$ is the desired $\epsilon_{j}(R)$ -splitting. Claim 6 gives

\displaystyle\fint_{B_{R}(x_{j})}|\nabla^{2}v^{\ell}_{j}|^{2}\to 0\,,

(3.111)

for all $1\leq\ell\leq k$ , while (3.109) and (3.110) imply

\displaystyle\fint_{B_{R}(x_{j})}|\langle\nabla v^{a}_{j},\nabla v^{b}_{j}\rangle-\delta^{ab}|\to 0\,.

(3.112)

To see that $v_{j}:B_{1}(x_{j})\to\mathds{R}^{k}$ is an $\epsilon_{j}(R)$ -splitting on $B_{1}(x_{j})$ , the last step is to show that $|\nabla v^{k}_{j}|\leq 1+\epsilon_{j}\to 1$ . However this follows immediately from (3.111) and (3.112) by using precisely the same argument as in (3.30)–(3.34).

Thus, for $j$ sufficiently large we see that $v_{j}:B_{1}(x_{j})\to\mathds{R}^{k}$ is an $\epsilon$ -splitting. This is a contradiction, so the proof is complete.

∎

[01YB]

4. Proof of Theorem 1.8, the Slicing Theorem

It is the goal of this Section to prove the Slicing Theorem (Theorem 1.8). Recall the statement:

For each $\epsilon>0$ there exists $\delta(n,\epsilon)>0$ such that if $M^{n}$ satisfies ${\rm Ric}_{M^{n}}\geq-(n-1)\delta$ and if $u:B_{2}(p)\to\mathds{R}^{n-2}$ is a harmonic $\delta$ -splitting map, then there exists a subset $G_{\epsilon}\subseteq B_{1}(0^{n-2})$ which satisfies the following:

(1)

${\rm Vol}(G_{\epsilon})>{\rm Vol}(B_{1}(0^{n-2}))-\epsilon$ .
(2)

If $s\in G_{\epsilon}$ then $u^{-1}(s)$ is nonempty.
(3)

For each $x\in u^{-1}(G_{\epsilon})$ and $r\leq 1$ there exists a lower triangular matrix $A\in GL(n-2)$ such that $A\circ u:B_{r}(x)\to\mathds{R}^{n-2}$ is an $\epsilon$ -splitting map.

The main tool will be the Transformation theorem (Theorem 1.11) proved in Section 3 and its applications. Recall first from Definition 1.10 the singular radius $s_{x}^{\eta}$ , where $\eta(n,\epsilon)$ such that for $\delta<\eta^{2}$ the conclusions of Theorem 1.11 hold for $\epsilon>0$ . It is clear from the definition that if $s^{\eta}_{x}>0$ , then there exists $\ell=\ell_{x}$ such that

(s^{\eta}_{x})^{2}\fint_{B_{s^{\eta}_{x}}(x)}|\Delta|\omega^{\ell}|\,|=\eta\fint_{B_{s^{\eta}_{x}}(x)}|\omega^{\ell}|\geq\frac{1}{2}\eta\fint_{B_{s^{\eta}_{x}}(x)}|\omega|\,,

(4.1)

where in the second inequality, we have used that $u$ is an $\epsilon$ -splitting to conclude that $|\omega|\leq 2|\omega^{\ell}|$ , for $\epsilon\leq\epsilon(n)$ sufficiently small.

Put

\mathcal{B}_{\eta}=:\,\bigcup_{x\,|\,s^{\eta}_{x}>0}B_{s^{\eta}_{x}}(x)\,.

Let $|u(B_{r}(x))|$ denote the $(n-2)$ -dimensional measure of the image $u(B_{r}(x))$ . In view of the Transformation theorem, to conclude the proof of the Slicing theorem, it suffices to show

|u(\mathcal{B}_{\eta})|\leq\delta^{\prime}(n,\delta)\,,

(4.2)

where $\delta^{\prime}(n,\delta)\to 0$ as $\delta\to 0$ .

Let us denote by $\mu$ , the measure such that

\displaystyle d\mu=\left(\int_{B_{2}(p)}|\omega|\right)^{-1}\cdot|\omega|dv_{g}\,.

(4.3)

Note that $\mu$ is a probability measure on $B_{2}(p)$ . In the arguments to come we will need $\mu$ have a certain doubling property. While in principle, it is too much to ask that $\mu$ is actually a doubling measure, next we observe that $\mu$ has a partial doubling property which will suffice for our purposes.

[01YC]

Lemma 4.1.

For each $x$ and $1/2\geq r\geq s^{\eta}_{x}$ we have the doubling condition

\mu(B_{2r}(x))\leq C(n)\mu(B_{r}(x))\,.

(4.4)

[01YD]

Proof.

By Theorem 1.11, there exists a lower triangular matrix $A\in GL(n-2)$ such that

\displaystyle u^{\prime}=A\circ u:B_{2r}(x)\to\mathds{R}^{n-2}

(4.5)

is an $\epsilon$ -splitting. Let $dv_{g}$ denote the Riemannian measure and set $\omega^{\prime}\equiv du^{\prime 1}\wedge\cdots\wedge du^{\prime n-2}$ . Define the measure $\mu^{\prime}$ by $\mu^{\prime}=\Big(\int_{B_{2}(p)}|\omega|\Big)^{-1}|\omega^{\prime}|dv_{g}$ . Then

\displaystyle\mu^{\prime}=\det(A)\mu\,.

(4.6)

In particular this gives us

\displaystyle\frac{\mu^{\prime}(B_{2r}(x))}{\mu^{\prime}(B_{r}(x))}=\frac{\mu(B_{2r}(x))}{\mu(B_{r}(x))}\,,

(4.7)

and it is equivalent to show the ratio bound for $\mu^{\prime}$ . Now since $u^{\prime}$ is an $\epsilon$ -splitting we have the estimate

\displaystyle\fint_{B_{2r}(x)}|\,|\omega^{\prime}|-1|\leq C(n)\epsilon\,.

(4.8)

Hence, we also have the estimate

\displaystyle\fint_{B_{r}(x)}|\,|\omega^{\prime}|-1|\leq\frac{{\rm Vol}(B_{2r}(x))}{{\rm Vol}(B_{r}(x))}\fint_{B_{2r}(x)}|\,|\omega^{\prime}|-1|\leq C(n)\epsilon\,,

(4.9)

which of course uses the doubling property for the Riemannian measure. By combining the previous two estimates we get

	$\displaystyle\big(1-C\epsilon\big){\rm Vol}(B_{r}(x))$	$\displaystyle\leq\mu^{\prime}(B_{r}(x))\leq\big(1+C\epsilon\big){\rm Vol}(B_{r}(x))\,$
	$\displaystyle\big(1-C\epsilon\big){\rm Vol}(B_{2r}(x))$	$\displaystyle\leq\mu^{\prime}(B_{2r}(x))\leq\big(1+C\epsilon\big){\rm Vol}(B_{2r}(x))\,.$		(4.10)

Finally, by using the definition of $\mu^{\prime}$ we arrive at:

$\displaystyle\mu^{\prime}(B_{2r}(x))$	$\displaystyle=\Big(\int_{B_{2}(p)}\|\omega\|\,dv_{g}\Big)^{-1}\int_{B_{2r}(x)}\|\omega^{\prime}\|$
	$\displaystyle\leq(1+C(n)\epsilon)\Big(\int_{B_{2}(p)}\|\omega\|\,dv_{g}\Big)^{-1}{\rm Vol}(B_{2r}(x))$
	$\displaystyle\leq C(n)\Big(\int_{B_{2}(p)}\|\omega\|\,dv_{g}\Big)^{-1}{\rm Vol}(B_{r}(x))$	(4.11)
	$\displaystyle\leq C(n)\Big(\int_{B_{2}(p)}\|\omega\|\,dv_{g}\Big)^{-1}\int_{B_{r}(x)}\|\omega^{\prime}\|$
	$\displaystyle=C(n)\mu^{\prime}(B_{r}(x))\,,$	(4.12)

which by (4.7) completes the proof. ∎

By a standard covering lemma, let us choose a collection of disjoint balls, $\{B_{s_{j}}(x_{j})\}\equiv\{B_{s^{\eta}_{j}}(x_{j})\}$ such that

\mathcal{B}_{\eta}\subset\bigcup_{j}B_{6s_{j}}(x_{j})\,.

(4.13)

For each such ball $B_{s_{j}}(x_{j})$ let $1\leq\ell_{j}\leq n-2$ be such that

$\displaystyle(s_{j})^{2}\fint_{B_{s_{j}}(x_{j})}\|\Delta\|\omega^{\ell_{j}}\|\|$	$\displaystyle=\eta\fint_{B_{s_{j}}(x_{j})}\|\omega^{\ell_{j}}\|\geq\frac{1}{2}\eta\fint_{B_{s_{j}}(x_{j})}\|\omega\|$
	$\displaystyle\geq\frac{1}{2}\frac{{\rm Vol}(B_{1}(x_{j}))}{{\rm Vol}(B_{s_{j}}(x_{j}))}\frac{\int_{B_{2}(p)}\|\omega\|}{{\rm Vol}(B_{1}(x_{j}))}\eta\Big(\int_{B_{2}(p)}\|\omega\|\Big)^{-1}\int_{B_{s_{j}}(x_{j})}\|\omega\|\,,$
	$\displaystyle\geq C(n)^{-1}\eta\,\mu(B_{s_{j}}(x_{j}))\,.$	(4.14)

Now since the balls $\{B_{s_{j}}(x_{j})\}$ are mutually disjoint, Theorem 1.9, together with (4.14) and Lemma 4.1 (the doubling property of $\mu$ ) gives

$\displaystyle\sum({6s_{j}})^{-2}\mu(B_{6s_{j}}(x_{j}))$	$\displaystyle\leq C(n)\sum s_{j}^{-2}\mu(B_{s_{j}}(x_{j}))$
	$\displaystyle\leq C(n)\sum\int_{B_{s_{j}}(x_{j})}\|\Delta\|\omega^{\ell_{j}}\|\,\|$	(4.15)
	$\displaystyle\leq C(n)\eta^{-1}\int_{B_{3/2}(p)}\|\Delta\|\omega^{\ell_{j}}\|\,\|\leq\delta^{\prime}(n,\delta)\,.$	(4.16)

The proof of the Slicing theorem (Theorem 1.8) requires that the image of $\mathcal{B}_{\delta}$ under $u$ have small measure. If in (4.15) the measure $\mu$ were instead the usual riemannian measure, then since $u$ is Lipschitz, standard estimates could be used to show just that. On the face of it, however, the $\mu$ -content estimate is much weaker, since for balls where the determinant $|\omega|$ of $u$ is small we have $\mu(B_{r}(x))<<{\rm Vol}(B_{r}(x))$ .

On the other hand, in the spirit of Sard’s theorem, we will see in the next lemma that at least for balls $B_{r}(x)$ with $r\geq s^{\delta}_{x}$ , we recover this loss because the volume of the image $u(B_{r}(x))$ is correspondingly small.

[01YE]

Lemma 4.2.

If $1/2\geq r\geq s^{\eta}_{x}$ , then

|u(B_{r}(x)|\leq C(n)\cdot r^{-2}\mu(B_{r}(x))\,.

(4.17)

[01YF]

Proof.

As in Lemma 4.1, choose a lower triangular matrix $A\in GL(n-2)$ such that

\displaystyle u^{\prime}=A\circ u:B_{2r}(x)\to\mathds{R}^{n-2}

(4.18)

is an $\epsilon$ -splitting and define the measure $\mu^{\prime}$ as in Lemma 4.1. Then as in (4.5), $\mu^{\prime}=\det(A)\mu$ .

Since $u^{\prime}$ is an $\epsilon$ -splitting, we have the estimates

		$\displaystyle\fint_{B_{2r}(x)}\|\|\omega^{\prime}\|-1\|\leq C(n)\epsilon\,,$
		$\displaystyle u^{\prime}(B_{r}(x))\subseteq B_{2r}(u^{\prime}(x))\,.$		(4.19)

By the first estimate above,

$\displaystyle\mu^{\prime}(B_{r}(x))$	$\displaystyle=\Big(\int_{B_{2}(p)}\|\omega\|\Big)^{-1}\int_{B_{r}(x)}\|\omega^{\prime}\|\,,$
	$\displaystyle\geq(1-C(n)\epsilon)\frac{{\rm Vol}(B_{r}(x))}{{\rm Vol}(B_{2}(p))}\fint_{B_{r}(x)}\|\omega^{\prime}\|$
	$\displaystyle\geq(1-C\epsilon)\frac{{\rm Vol}(B_{r}(x))}{{\rm Vol}(B_{3}(x))}\geq C(n)r^{n}\,,$	(4.20)

where in the last step we have used volume monotonicity for the Riemannian measure. On the other hand, by the second estimate of (4.19),

\displaystyle|u^{\prime}(B_{r}(x))|\leq C(n)r^{n-2}\,.

(4.21)

Combining these gives the estimate

\displaystyle|u^{\prime}(B_{r}(x))|\leq C(n)r^{-2}\mu^{\prime}(B_{r}(x))\,.

(4.22)

To relate these back to the original function $u$ , we observe that

	$\displaystyle\|u^{\prime}(B_{r}(x))\|$	$\displaystyle=\det(A)\|u(B_{r}(x))\|\,,$
	$\displaystyle\mu^{\prime}(B_{r}(x))$	$\displaystyle=\det(A)\|\mu(B_{r}(x))\|\,,$		(4.23)

which immediately gives

\displaystyle|u(B_{r}(x))|\leq C(n)r^{-2}\mu(B_{r}(x))\,.

(4.24)

This completes the proof. ∎

We can now finish the proof of Theorem 1.8. Indeed, we have by (4.13), (4.15), (4.17), that

\displaystyle|u(\mathcal{B}_{\eta})|\leq\sum|u(B_{6s_{j}}(x_{j}))|\leq\sum s_{j}^{-2}\mu(B_{6s_{j}}(x_{j}))\leq\delta^{\prime}(n,\delta)\,.

(4.25)

By taking $\delta$ sufficiently small, this suffices to complete the proof.

[01YG]

5. Codimension $4$ Regularity of Singular Limits

In this section we prove Theorem 1.1. Thus, we consider a Gromov-Hausdorff limit space,

\displaystyle(M^{n}_{j},d_{j},p_{j})\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}(X,d,p)\,,

(5.1)

of a sequence of Riemannian manifolds $(M^{n}_{j},g_{j},p_{j})$ , satisfying $|{\rm Ric}_{M^{n}_{j}}|\leq n-1$ and ${\rm Vol}(B_{1}(p_{j}))>{\rm v}>0$ . We will show that there exists a subset $\mathcal{S}\subseteq X$ of codimension $4$ such that $X\setminus\mathcal{S}$ is a $C^{1,\alpha}$ -Riemannian manifold. In this section, we will show that $\mathcal{S}$ has Hausdorff codimension 4. We will postpone the improvement to Minkowski codimension $4$ until Section 7.

As mentioned in Section 1, it has been understood since [ChCo2] that the main technical challenge lies in showing that spaces of the form $\mathds{R}^{n-2}\times C(S^{1}_{\beta})$ , where $S^{1}_{\beta}$ is the circle of circumference $\beta\leq 2\pi$ , cannot arise as limit spaces unless $\beta=2\pi$ and hence $\mathds{R}^{n-2}\times C(S^{1}_{\beta})=\mathds{R}^{n}$ . The Slicing Theorem (Theorem 1.8) was expressly designed to enable us to handle this point via a blow up argument. We will do this in Section 5.1.

In subsection 5.2 we then prove that more general spaces of the form $\mathds{R}^{n-3}\times C(Y)$ cannot arise as limit spaces. The proof of this statement, has a very different feel than the proof ruling out the codimension two limits, and essentially comes down to a bordism and curvature pinching argument for $3$ -manifolds.

Finally, in Section 5.3 we combine the tools developed in the previous subsections to prove the Hausdorff estimates of Theorem 1.1.

[01YH]

5.1. Nonexistence of Codimension $2$ Singularities

In this subsection, we use the tools of Section 4 in order to prove that spaces that are $(n-2)$ -symmetric cannot arise as noncollapsed limits of manifolds with bounded Ricci curvature.

[01YI]

Theorem 5.1 ( $(n-2)$ -Symmetric Limits).

Let $(M^{n}_{j},g_{j},p_{j})$ be a sequence of Riemannian manifolds satisfying $|{\rm Ric}_{M^{n}_{j}}|\to 0$ , ${\rm Vol}(B_{1}(p_{j}))>{\rm v}>0$ and such that

\displaystyle(M_{j}^{n},d_{j},p_{j})\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}\mathds{R}^{n-2}\times C(S^{1}_{\beta})\,.

(5.2)

Then $\beta=2\pi$ and $\mathds{R}^{n-2}\times C(S^{1}_{\beta})=\mathds{R}^{n}$ .

[01YJ]

Proof of Theorem 5.1.

We will prove the result by contradiction. So let us assume it is false. Then there exists a sequence $(M^{n}_{j},g_{j},p_{j})$ of Riemannian manifolds satisfying $|{\rm Ric}_{M^{n}_{j}}|\to 0$ , ${\rm Vol}(B_{1}(p_{j}))>{\rm v}>0$ and such that

\displaystyle(M_{j}^{n},d_{j},p_{j})\to\big(\mathds{R}^{n-2}\times C(S^{1}_{\beta}),d,p\big)\,,

(5.3)

with $\beta<2\pi$ and $p$ a vertex.

Note first that by the noncollapsing assumption we have $\beta\geq\beta_{0}(n,v)$ .

Now by Lemma 1.7, there exists $\delta_{j}$ -splitting maps $u_{j}:B_{2}(p_{j})\to\mathds{R}^{n-2}$ with $\delta_{j}\to 0$ . Fix some sequence $\epsilon_{j}\to 0$ which is tending to zero so slowly compared to $\delta_{j}$ , that Theorem 1.8 holds for $u_{j}:B_{2}(0)\to\mathds{R}^{n-2}$ with $\epsilon_{j}$ . Let $G_{\epsilon_{j}}\subseteq B_{1}(0^{n-2})$ be the corresponding good values of $u_{j}$ , and let $s_{j}\in G_{\epsilon_{j}}\cap B_{10^{-1}}(0^{n-2})$ be fixed regular values.

Note that $\mathds{R}^{n-2}\times C(S^{1}_{\beta})$ is smooth outside of the singular set $\mathcal{S}=\mathds{R}^{n-2}\times\{0\}\subseteq\mathds{R}^{n-2}\times C(S^{1}_{\beta})$ . In particular on $\mathds{R}^{n-2}\times C(S^{1}_{\beta})$ we have $r_{h}(x)\approx 1/d(x,\mathcal{S})$ , where $r_{h}$ is the harmonic radius as in Section 1 and $d$ denotes distance. By the standard $\epsilon$ -regularity theorem, it follows that the convergence of $M^{n}_{j}$ is in $C^{1,\alpha}\cap W^{2,q}$ away from $\mathcal{S}$ , for every $\alpha<1$ and $q<\infty$ . Let $f_{j}:B_{\epsilon^{-1}_{j}}(p)\to B_{\epsilon^{-1}_{j}}(p_{j})$ be the $\epsilon_{j}$ -Gromov Hausdorff maps, and let us denote $\mathcal{S}_{j}\equiv f_{j}(\mathcal{S})\subseteq M^{n}_{j}$ . Then by the previous statements, for every $\tau>0$ , all $j$ sufficiently large, and $x\in B_{1}(p_{j})\setminus T_{\tau}(\mathcal{S}_{j})$ , we have $r_{h}(x)\geq\frac{\tau}{2}$ .

Consider again the submanifold $u^{-1}_{j}(s_{j})\cap B_{1}(p_{j})$ . Define the scale

\displaystyle r_{j}=\min\{r_{h}(x):x\in u^{-1}_{j}(s_{j})\cap B_{1}(p_{j})\}\,.

(5.4)

By the considerations of the previous paragraph, this minimum is actually obtained at some $x_{j}\in u^{-1}_{j}(s_{j})\cap B_{1}(p_{j})$ , with $x_{j}\to\mathcal{S}_{j}\cap B_{10^{-1}}(p_{j})$ . Moreover, since $S^{1}_{\beta}$ , the cross-section of the cone factor, satisfies $0<\beta<2\pi$ , it follows that $r_{j}\to 0$ . According to Theorem 1.8, there exists a lower triangular matrix $A_{j}\in GL(n-2)$ such that $v_{j}\equiv A_{j}\circ\big(u_{j}-s_{j}\big):B_{r_{j}}(x_{j})\to\mathds{R}^{n-2}$ is an $\epsilon_{j}$ -splitting map. Note that we have renormalized so that each of our regular values is the zero level set.

Now let us consider the sequence $(M^{n}_{j},r_{j}^{-1}d_{j},x_{j})$ . After passing to a subsequence if necessary, which we will continue to denote by $(M^{n}_{j},r_{j}^{-1}d_{j},x_{j})$ , have

\displaystyle(M^{n}_{j},r_{j}^{-1}d_{j},x_{j})\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}(X,d_{X},x)\,,

(5.5)

in the pointed Gromov-Hausdorff sense, where $X$ splits off $\mathds{R}^{n-2}$ isometrically.

We begin by observing that by our noncollapsing assumption we have ${\rm Vol}(B_{1}(x_{j}))>c(n){\rm v}>0$ , and hence, in the rescaled spaces, we have ${\rm Vol}(B_{r}(x_{j}))>c{\rm v}r^{n}$ for all $r\leq R_{j}\to\infty$ . In particular, $X$ has Euclidean volume growth at $\infty$ i.e. ${\rm Vol}(B_{r}(x^{\prime}))>c{\rm v}\,r^{n}$ for all $r>0$ .

After possibly passing to another subsequence, we can limit the functions $v_{j}$ to a function $v:X\to\mathds{R}^{n-2}$ . Note that by our normalization, we have $v_{j}:B_{2}(x_{j})\to\mathds{R}^{n-2}$ are $\epsilon_{j}$ -splittings, and that by Theorem 1.11, we have for each $R>2$ that $v_{j}:B_{R}(x_{j})\to\mathds{R}^{n-2}$ are $C(n,R)\epsilon_{j}$ -splittings. In particular, we can conclude that

\displaystyle X=\mathds{R}^{n-2}\times S\,,

(5.6)

where $v:X\to\mathds{R}^{n-2}$ is the projection map and $S=u^{-1}(0)$ .

Now by construction, in the rescaled spaces we have for any $y\in u^{-1}_{j}(0)$ that $r_{h}(y)\geq 1$ . Therefore, the limit $X$ is $C^{1,\alpha}\cap W^{2,q}$ in a neighborhood of $u^{-1}(0)$ , and hence $S=u^{-1}(0)$ is a nonsingular surface. Thus, since $X=\mathds{R}^{n-2}\times S$ it follows that $X$ is at least a $C^{1,\alpha}\cap W^{2,q}$ manifold with $r_{h}\geq 1$ . Since the Ricci curvature is uniformly bounded, in fact tending to zero, we have by the standard $\epsilon$ -regularity theorem that the convergence $(M^{n}_{j},r_{j}^{-1}d_{j},x_{j})\to(X,d_{X},x)$ is in $C^{1,\alpha}\cap W^{2,q}$ . Because the convergence is in $C^{1,\alpha}\cap W^{2,q}$ we have that $r_{h}$ converges continuously; [A90]. In particular, we have that $r_{h}(x^{\prime}_{j})\to r_{h}(x^{\prime})$ and so $r_{h}(x^{\prime})=1$ .

On the other hand, since $|{\rm Ric}_{M^{n}_{j}}|\to 0$ and $X$ is $C^{1,\alpha}\cap W^{2,q}$ it follows that $X$ is a smooth Ricci flat manifold. This is easiest to see by writing directly in harmonic coordinates on $X$ , see [A90] for the argument. Now since $X=\mathds{R}^{n-2}\times S$ , we can conclude that $S$ is smooth and Ricci flat, hence flat. In particular, we have that $X$ is flat. Since we have already shown that $X$ has Euclidean volume growth, this implies that $X=\mathds{R}^{n}$ . However, we have also already concluded that $r_{h}(x^{\prime})=1$ , which gives us our desired contradiction. ∎

We end this subsection with the following corollary, which states that a noncollapsed limit space is smooth away from a set of codimension $3$ . We will use this in the next subsection to show $(n-3)$ -symmetric splittings cannot arise as limits.

[01YK]

Corollary 5.2.

Let $(M^{n}_{j},g_{j},p_{j})$ be a sequence of Riemannian manifolds satisfying $|{\rm Ric}_{M^{n}_{j}}|\leq n-1$ , ${\rm Vol}(B_{1}(p_{j}))>{\rm v}>0$ and such that

\displaystyle(M_{j}^{n},d_{j},p_{j})\to(X,d,p)\,.

(5.7)

Then there exists a subset, $\mathcal{S}\subseteq X$ , with $\dim\mathcal{S}\leq n-3$ , such that for each $x\in X\setminus\mathcal{S}$ , we have $r_{h}(x)>0$ . In particular, $x\in X\setminus\mathcal{S}$ is a $C^{1,\alpha}$ Riemannian manifold.

[01YL]

Proof.

Recall the standard stratification of $X$ . In particular, if we consider the subset $\mathcal{S}^{n-3}\subset X$ we have that $\dim\mathcal{S}^{n-3}\leq n-3$ , and that for every point $x\not\in\mathcal{S}^{n-3}$ there exists some tangent cone at $x$ which is isometric to $\mathds{R}^{n-2}\times C(S^{1}_{\beta})$ . That is, there exists $r_{a}\to 0$ such that

\displaystyle(X,r_{a}^{-1}d,x)\to\mathds{R}^{n-2}\times C(S^{1}_{\beta})\,.

(5.8)

However by Theorem 5.1 we then have $\beta=2\pi$ , which is to say that

\displaystyle(X,r_{a}^{-1}d,x)\to\mathds{R}^{n}\,.

(5.9)

Thus, for $a\in\mathds{N}$ sufficiently large, we can apply the standard $\epsilon$ -regularity theorem, Theorem 2.3, to see that a neighborhood of $x$ is a $C^{1,\alpha}$ Riemannian manifold, which proves the corollary. ∎

[01YM]

5.2. Nonexistence of Codimension $3$ singularities

In this subsection we use the tools of Section 4 and Section 5.1 in order to prove that $(n-3)$ -symmetric metric spaces cannot arise as limits of manifolds with bounded Ricci curvature. Specifically, we prove the following:

[01YN]

Theorem 5.3 ( $(n-3)$ -Symmetric Limits).

Let $(M^{n}_{j},g_{j},p_{j})$ be a sequence of Riemannian manifolds satisfying $|{\rm Ric}_{M^{n}_{j}}|\to 0$ , ${\rm Vol}(B_{1}(p_{j}))>{\rm v}>0$ and such that

\displaystyle(M_{j}^{n},d_{j},p_{j})\to\mathds{R}^{n-3}\times C(Y)\,,

(5.10)

in the pointed Gromov-Hausdorff sense, where $Y$ is some compact metric space. Then $Y=S^{2}(1)$ is isometric to the unit $2$ -sphere and hence $\mathds{R}^{n-3}\times C(Y)=\mathds{R}^{n}$ .

[01YP]

Proof.

Let us assume that this is not the case and study such a limit space $\mathds{R}^{n-3}\times C(Y)$ . The first observation is that by Corollary 5.2, it follows that $Y$ is a smooth surface. Indeed, if there were a point $y\in Y$ such that $r_{h}(y)=0$ . Then since $X=\mathds{R}^{n-3}\times C(Y)$ it would follow that there is a set of codimension at least $2$ such that $r_{h}\equiv 0$ , which cannot happen by Corollary 5.2.

Since $Y$ a $C^{1,\alpha}\cap W^{2,q}$ manifold and $|{\rm Ric}_{M^{n}_{j}}|\to 0$ , it follows that $Y$ is a smooth Einstein manifold satisfying ${\rm Ric}_{Y}=g$ . Because $Y$ is a surface, this means in particular that $Y$ has constant sectional curvature $\equiv 1$ . Thus, either $Y=\mathds{R}\mathds{P}^{2}$ or $Y=S^{2}$ , the unit $2$ -sphere, and in the latter case we are done.

So let us study the case $Y=\mathds{R}\mathds{P}^{2}$ . For $\epsilon>0$ small, choose $u_{j}:B_{2}(p_{j})\to\mathds{R}^{n-3}$ to be an $\epsilon$ -splitting as in Lemma 1.7. Note that away from the singular set $\mathcal{S}\equiv\mathds{R}^{n-3}\times\{0\}$ we have that the $M_{j}^{n}$ converge to $\mathds{R}^{n-3}\times C(Y)$ in $C^{1,\alpha}$ . If $f_{j}:B_{2}(p)\to B_{2}(p_{j})$ denote the Gromov-Hausdorff maps, we put $\mathcal{S}_{j}=f_{j}(\mathcal{S})$ . Then for $\tau>0$ small but fixed, we have for $j$ sufficiently large, that on $B_{1}(p)\setminus T_{\tau}(\mathcal{S}_{j})$ , the estimates $|\nabla u_{j}|>\frac{1}{2}$ and $|\nabla^{2}u_{j}|\leq 1$ hold.

Consider Poisson approximation $h_{j}$ to the square of distance function $d^{2}(x,p_{j})$ on $B_{2}(p_{j})$ . That is, $\Delta h_{j}=2n$ and $h_{j}=1$ on $\partial B_{2}(p_{j})$ . We have (see for instance [ChCo1]) that $|h_{j}-d(\cdot,p_{j})|\to 0$ uniformly in $B_{2}(p_{j})$ , and again because the convergence is in $C^{1,\alpha}$ we have for $j$ sufficiently large that $|\nabla h|>\delta$ and $|\nabla^{2}h|\leq 4n$ on $B_{1}(p)\setminus B_{\tau}(\mathcal{S}_{j})$ . Once again, appealing to the $C^{1,\alpha}$ convergence, for all $j$ sufficiently large and all $s\in B_{1}(0^{n-3})$ we have that $u^{-1}(s)\cap h^{-1}(1)$ is diffeomorphic to $\mathds{R}\mathds{P}^{2}$ . By Sard’s theorem, there exists a regular value $s_{j}\in B_{1}(0^{n-3})$ . Then for $j$ sufficiently large, $u^{-1}_{j}(s_{j})\cap\{h\leq 1\}$ is a smooth $3$ -manifold, whose boundary is diffeomorphic to $\mathds{R}\mathds{P}^{2}$ . However, the second Stiefel-Whitney number of $\mathds{R}\mathds{P}^{2}$ is nonzero, and in particular, $\mathds{R}\mathds{P}^{2}$ does not bound a smooth $3$ -manifold. This contradicts $Y=\mathds{R}\mathds{P}^{2}$ .

∎

[01YQ]

5.3. Proof of Hausdorff Estimates of Theorem 1.1

With Theorem 5.3 in hand, the proof of Theorem 1.1 becomes standard, and follows the same lines as the proof of Corollary 5.2. Thus, consider a sequence

\displaystyle(M^{n}_{j},d_{j},p_{j})\to(X,d,p)

(5.11)

of Riemannian manifolds satisfying $|{\rm Ric}_{M^{n}_{j}}|\leq n-1$ and ${\rm Vol}(B_{1}(p_{j}))>{\rm v}>0$ , which Gromov-Hausdorff converge to some $X$ . Recall again the standard stratification of $X$ , which is reviewed in Section 2.1. More specifically let us consider the closed stratum $\mathcal{S}^{n-4}(X)\subseteq X$ . On the one hand, we have from [ChCo1]

\displaystyle\dim\mathcal{S}^{n-4}\leq n-4\,.

(5.12)

On the other hand, we have that for every point $x\not\in\mathcal{S}^{n-4}$ , there exists some tangent cone at $x$ which is isometric to $\mathds{R}^{n-3}\times C(Y)$ . That is, for some sequence $r_{a}\to 0$ we have

\displaystyle(X,r_{a}^{-1}d,x)\to\mathds{R}^{n-3}\times C(Y)\,.

(5.13)

However, by Theorem 5.3, we have that $Y$ is isometric to the unit $2$ -sphere, and hence,

\displaystyle(X,r_{a}^{-1}d,x)\to\mathds{R}^{n}\,.

(5.14)

Then for $a\in\mathds{N}$ sufficiently large, we can apply the standard $\epsilon$ -regularity theorem, Theorem 2.3, to see that $r_{h}(x)>0$ , and hence that a neighborhood of $x$ is a $C^{1,\alpha}$ Riemannian manifold, which proves the theorem.

[01YR]

6. The $\epsilon$ -regularity Theorem

In Section 5, we showed that limit spaces satisfying our assumptions must be smooth away from a closed subset of codimension $4$ . However, the strongest applications come from a more effective version of this statement. In particular, the curvature estimates of Theorem 1.3 and the Minkowski estimates of Theorem 1.1 will require a more rigid statement. Namely, we will prove the following in this section:

[01YS]

Theorem 6.1.

There exists $\epsilon(n,{\rm v})>0$ such that if $M^{n}$ satisfies $|{\rm Ric}_{M^{n}}|\leq\epsilon$ , ${\rm Vol}(B_{1}(p))>{\rm v}>0$ . and

\displaystyle d_{GH}\big(B_{2}(p),B_{2}(0)\big)<\epsilon\,,

(6.1)

where $0$ is a vertex of the cone $\mathds{R}^{n-3}\times C(Y)$ , for some metric space $Y$ , then we have

\displaystyle r_{h}(p)\geq 1\,.

(6.2)

Consequently, if $M^{n}$ is Einstein, we have the bound

\displaystyle\sup_{B_{1}(p)}|{\rm Rm}|\leq 1\,.

(6.3)

[01YT]

Proof.

Given $n$ and ${\rm v}>0$ , assume no such $\epsilon$ exists. Then there exists a sequence of spaces $(M^{n}_{j},g_{j},p_{j})$ such that $|{\rm Ric}_{M^{n}_{i}}|\leq\epsilon_{j}\to 0$ , ${\rm Vol}(B_{1}(p_{j}))>{\rm v}>0$ and

\displaystyle d_{GH}\big(B_{2}(p_{j}),B_{2}(0_{j})\big)<\epsilon_{j}\to 0\,,

(6.4)

where $0_{j}\in\mathds{R}^{n-3}\times C(Y_{j})$ is a vertex but $r_{h}(p)<1$ . After possibly passing to a subsequence,we have

\displaystyle B_{2}(p_{j})\to B_{2}(0)\,,

(6.5)

where $0\in\mathds{R}^{n-3}\times C(Y)\equiv X$ is a vertex. But if $C(Y)$ has any point with $r_{h}(x)=0$ , then there is a set of Hausdorff codimension $3$ in $X$ which is not smooth. By the Hausdorff estimate of Theorem 1.1 this is not possible, so we must have that $C(Y)$ is smooth. Thus, $Y$ is a smooth manifold, and in fact, $C(Y)$ is itself be smooth if and only if $Y$ is the unit $2$ -sphere. Thus,

\displaystyle B_{2}(p_{j})\to B_{2}(0^{n})\subseteq\mathds{R}^{n}\,.

(6.6)

But now, we can apply the standard $\epsilon$ -regularity theorem, to conclude $r_{h}(p)\leq 1$ , which is a contradiction. ∎

[01YU]

7. Quantitative Stratification and Effective Estimates

Having shown in Sections 5 and 6 that noncollapsed limits of Einstein manifolds are smooth away from a closed codimension $4$ subset, we will now give some applications. In particular, we will use the ideas of quantiative stratification first introduced in [ChNa13] in order to improve the codimension estimates on singular sets of limit spaces to curvature estimates on Einstein manifolds. More precisely, in this section, we will prove Theorem 1.3. We will also improve the Hausdorff dimension estimate of Theorem 1.1 to a Minkowski dimension estimate. One can view this as an easy corollary of Theorem 1.3.

We begin here by reviewing the quantitative stratification and the main results on it from [ChNa13]. These will play a crucial role in our estimates. In subsection 7.1 we combine the main results concerning the quantitative stratification, stated in Theorem 7.3, with the $\epsilon$ -regularity of Theorem 6.1 in order to prove the main estimates on Einstein manifolds given in Theorem 1.3. In subsection 7.2 we apply the regularity results of Theorem 1.3 in order to conclude stronger results about the behavior of harmonic functions on Einstein manifolds.

The idea of [ChNa13] was to make the notion of stratification more effective. The standard stratification, recalled in Section 2.1, is used to show that that most points have a lot of symmetry infinitesimally. The quantitative stratification is used to show that most balls of a definite size have a lot of approximate symmetry. In particular, the quantitative stratification introduced in [ChNa13] exists and gives nontrivial information even on a smooth manifold, unlike the standard stratification which is always trivial on a smooth space. This point is crucial to the proof of Theorem 1.3. To make this precise we begin by defining a more local version of approximate symmetry.

[01YV]

Definition 7.1.

Given a metric space $Y$ with $y\in Y$ , $r>0$ and $\epsilon>0$ , we say that $y$ is $(k,\epsilon,r)$ -symmetric if there exists a $k$ -symmetric space $Y^{\prime}$ such that $d_{GH}(B_{r}(y),B_{r}(y^{\prime}))<\epsilon r$ , where $y^{\prime}\in Y^{\prime}$ is a vertex.

Recall from Section that $Y^{\prime}$ is $k$ -symmetric if $Y^{\prime}=\mathds{R}^{k}\times C(Z^{\prime})$ . To state the definition in words, we say that $Y$ is $(k,\epsilon,r)$ -symmetric if the ball $B_{r}(x)$ looks very close to having $k$ -symmetries. The quantitative stratification is then defined as follows:

[01YW]

Definition 7.2.

For each $\epsilon,r>0$ and $k\in\mathds{N}$ , define the closed quantitative $k$ -stratum, $S^{k}_{\epsilon,r}(X)$ , by

\displaystyle S^{k}_{\epsilon,r}(X)\equiv\{x\in X:\text{ for no $r\leq s\leq 1$ is $x$ a $(k,\epsilon,r)$-symmetric point}\}\,.

(7.1)

Thus, the closed stratum $S^{k}_{\epsilon,r}(X)$ is the collection of points such that no ball of size at least $r$ is almost $(k+1)$ -symmetric. The first main result of [ChNa13] is to show that for manifolds which are noncollapsed and have lower Ricci curvature bounds, the set $S^{k}_{\epsilon,r}(X)$ is small in a very strong sense. To say this a little more carefully, if one pretends that the $k$ -stratum is a well behaved $k$ -dimensional submanifold, then one would expect the volume of the $r$ -tube around the set to behave like $Cr^{n-k}$ . Although we don’t know this to be the case, the following slightly weaker statement does hold.

[01YX]

Theorem 7.3 (Quantitative Stratification,[ChNa13]).

Let $M^{n}$ satisfy ${\rm Ric}\geq-(n-1)$ with ${\rm Vol}(B_{1}(p))>v>0$ . Then for every $\epsilon,\eta>0$ there exists $C=C(n,{\rm v},\epsilon,\eta)$ such that

\displaystyle{\rm Vol}\left(T_{r}\left(\mathcal{S}^{k}_{\epsilon,r}(M)\cap B_{1}(p)\right)\right)\leq Cr^{n-k-\eta}\,.

(7.2)

[01YY]

7.1. Proof of Theorem 1.3

In this subsection we combine Theorem 6.1 and Theorem 7.3 in order to prove Theorem 1.3.

[01YZ]

Proof.

(of Theorem 1.3) Let $(M^{n},g,p)$ satisfy $|{\rm Ric}_{M^{n}}|\leq n-1$ and ${\rm Vol}(B_{1}(p))>{\rm v}>0$ . We will first show that for every $q<2$ there exists $C=C(n,{\rm v},q)>0$ such that

\displaystyle\fint_{B_{1}(p)}r_{h}^{-2q}\leq C\,.

(7.3)

Simultaneously, we will show that if $M^{n}$ is Einstein, then this can be improved to

\displaystyle\fint_{B_{1}(p)}r_{x}^{-2q}\leq C\,,

(7.4)

where $r_{x}$ denotes the regularity scale at $x$ .

Let $q<2$ and set $\eta=4-2q$ . Consider Theorem 7.3 with $\epsilon=\epsilon(n)>0$ chosen from Theorem 6.1 and $\eta$ as above. Thus, there exists $C(n,{\rm v},q)$ such that

\displaystyle{\rm Vol}(T_{r}(\{x\in\mathcal{S}^{n-4}_{\epsilon,2r}\cap B_{1}(p)\}))<Cr^{4-\eta}\,.

(7.5)

Note that by rescaling, we may regard the $\epsilon$ -regularity theorem (Theorem 6.1) as stating that if $x$ is $(n-3,\epsilon,2r)$ -symmetric then $r_{h}>r$ , and if $M^{n}$ is Einstein then $r_{x}>r$ . In fact, we have that if $x$ is $(n-3,\epsilon,s)$ -symmetric for any $s\geq 2r$ , then $r_{h}>r$ . This is to say that if $x\not\in\mathcal{S}^{n-4}_{\epsilon,2r}$ , then $r_{h}>\frac{s}{2}>r$ . The contrapositive gives the inclusion

\displaystyle\{x\in B_{1}(p):r_{h}\leq r\}\subseteq\mathcal{S}^{n-4}_{\epsilon,2r}\cap B_{1}(p)\,.

(7.6)

which by (7.5) gives us the desired estimate

\displaystyle{\rm Vol}(T_{r}(\{x\in B_{1}(p):r_{h}\leq r\}))<Cr^{4-\eta}\leq Cr^{2p}\,.

(7.7)

If $M^{n}$ is Einstein, then Theorem 6.1 allows us to replace $r_{h}$ with $r_{x}$ , as claimed.

Now, for $q<2$ , let us prove the $L^{q}$ bound on the curvature from Theorem 1.3. For this note that if $r_{h}(x)>r$ then by definition there exists harmonic coordinates $\Phi:B_{r}(0^{n})\to M$ with $\phi(0)=x$ and such that

\displaystyle||g_{ij}-\eta_{ij}||_{C^{0}(B_{r}(0))}+r||\partial_{k}g_{ij}||_{C^{0}(B_{r}(0))}<10^{-3}\,,

(7.8)

where $g_{ij}=\Phi^{*}g$ is the pullback metric. Since the Ricci curvature satisfies the bound $|{\rm Ric}_{M^{n}}|\leq n-1$ , this implies that

\displaystyle|\Delta_{x}g_{ij}|<C(n)r^{-2}\,,

(7.9)

where $\Delta_{x}$ denotes the Laplacian written in coordinates. In particular, for every $\alpha<1$ and $s<\infty$ , we have the scale invariant estimates

		$\displaystyle r^{1+\alpha}\|\|\partial_{k}g_{ij}\|\|_{C^{\alpha}(B_{\frac{3r}{4}}(0))}\leq C(n,\alpha)\,,$
		$\displaystyle r^{2}\|\|g_{ij}\|\|_{W^{2,s}(B_{\frac{3r}{4}}(0))}\leq C(n,s)\,.$		(7.10)

In particular, applying this to $s=q$ we get

\displaystyle r^{2q}\fint_{B_{r/2}(x)}|{\rm Rm}|^{q}\leq C(n)r^{2q}\fint_{B_{3r/4}(0)}|\Phi^{*}{\rm Rm}|^{q}<C(n,q)\,.

(7.11)

Let $\eta=2-q$ be chosen so that $q+\frac{\eta}{2}<2$ . Then we have already shown that

\displaystyle{\rm Vol}(T_{r}(\{x\in B_{1}(p):r_{h}\leq r\}))<Cr^{2q+\eta}\,,

(7.12)

for $C(n,{\rm v},q)>0$ . Consider the covering $\{B_{r_{h}(x)}(x)\}$ of $B_{1}(p)$ , and a subcovering $\{B_{r_{j}}(x_{j})\}$ by mutually disjoint balls, such that

(1)

$B_{1}(p)\subseteq\bigcup B_{r_{j}}(x_{j})$ with $r_{j}=\frac{1}{2}r_{h}(x)$ .
(2)

$\{B_{r_{j}/4}(x_{j})\}$ are disjoint.

By using (7.12), we see for each $\alpha\in\mathds{N}$ that

\displaystyle\sum_{2^{-\alpha-1}<r_{j}\leq 2^{-\alpha}}{\rm Vol}(B_{r_{j}}(x_{j}))\leq Cr_{j}^{2q+\eta}=C\,r_{j}^{2q}\,2^{-\eta\alpha}\,.

(7.13)

Summing over $\alpha$ this gives

\displaystyle\sum r_{j}^{-2q}{\rm Vol}(B_{r_{j}}(x_{j}))\leq C\sum 2^{-\eta\alpha}\leq C(n,{\rm v},q)\,.

(7.14)

Finally, combining this with (7.11) we get

	$\displaystyle\fint_{B_{1}(p)}\|{\rm Rm}\|^{q}$	$\displaystyle\leq C(n,{\rm v})\sum\int_{B_{r_{j}}(x_{j})}\|{\rm Rm}\|^{q}$
		$\displaystyle\leq C(n,{\rm v},q)\sum r_{j}^{-2q}{\rm Vol}(B_{r_{j}}(x_{j}))\leq C(n,{\rm v},q)\,,$		(7.15)

which finishes the proof of Theorem 1.3. ∎

[01Z0]

7.2. $L^{q}$ -Estimates for Harmonic Functions on Einstein Manifolds

In this subsection we give some applications of Theorem 1.3. In particular, we study Sobolev bounds of harmonic functions and solutions of more general equations on manifolds with bounded Ricci curvature. As we have used repeatedly, given a lower bound on Ricci curvature , there is a definite $L^{2}$ bound on the Hessian of a harmonic function; see (3.5). However, the example of a rounded off $2$ -dimensional cone shows that one does not have definite $L^{q}$ bounds for any $q>2$ ; see Example 2.1. In this subsection, we will see that the situation is better for noncollapsed spaces with bounded Ricci curvature. Namely, one can obtain $L^{q}$ bounds on the Hessians of such harmonic functions for all $q<4$ . More generally, we show the following:

[01Z1]

Theorem 7.4.

For every $q<4$ there exists $C=C(n,{\rm v},q)$ such that if $M^{n}$ satisfies $|{\rm Ric}_{M^{n}}|\leq n-1$ and ${\rm Vol}(B_{1}(p))>{\rm v}>0$ and $u:B_{2}(p)\to\mathds{R}$ satisfies

|u|\leq 1\,,

|\Delta u|\leq 1\,,

then for every $q<4$

\displaystyle\fint_{B_{1}(p)}|\nabla^{2}u|^{q}\leq C\,.

(7.16)

[01Z2]

Proof.

Note that by the Cheng-Yau gradient estimate, we have

\displaystyle\sup_{B_{3/2}(p)}|\nabla u|\leq C(n)\,.

(7.17)

Now using Theorem 1.3 we know for each $\epsilon>0$ that

\displaystyle{\rm Vol}(T_{r}(\{x\in B_{1}(p):r_{h}(x)\leq r\}))\leq C_{\epsilon}(n,{\rm v},\epsilon)r^{4-\epsilon}\,.

(7.18)

In particular, let us consider the sets

\displaystyle\mathcal{C}_{\alpha}\equiv\{x\in B_{1}(p):r_{\alpha}\leq r_{h}(x)\leq r_{\alpha-1}\}\,,

(7.19)

where $r_{\alpha}\equiv 2^{-\alpha}$ . For the set $\mathcal{C}_{\alpha}$ , we have the cover $\{B_{r_{\alpha}}(x)\}_{x\in\mathcal{C}_{\alpha}}$ . We can choose a finite subcovering $\{B_{r_{\alpha}/2}(x_{i})\}_{1}^{N_{\alpha}}$ such that the balls $B_{r_{\alpha}/8}(x_{i})$ are mutually disjoint. Using (7.18) we have

\displaystyle N_{\alpha}\leq C_{\epsilon}r_{\alpha}^{4-n-\epsilon}\,.

(7.20)

On each ball $B_{r_{\alpha}/2}(x_{j})$ we can use standard elliptic estimates along with the gradient bound $|\nabla u|\leq C(n)$ to get the scale-invariant estimate

\displaystyle r_{\alpha}^{q}\fint_{B_{r_{\alpha}/2}(x_{j})}|\nabla^{2}u|^{q}\leq C(n,{\rm v},q)\,,

(7.21)

for any $q<\infty$ . In particular, if we choose $q<4$ and pick $\epsilon=\frac{4-q}{2}$ , then we have

\displaystyle\int_{B_{r_{\alpha}/2}(x_{j})}|\nabla^{2}u|^{q}\leq C(n,{\rm v})\,r_{\alpha}^{n-4+2\epsilon}\,.

(7.22)

Combining this with (7.18) gives us

\displaystyle\int_{\mathcal{C}_{\alpha}}|\nabla^{2}u|^{q}\leq C(n,{\rm v})\,r_{\alpha}^{n-4+2\epsilon}\cdot N_{\alpha}\leq C(n,{\rm v},q)\,r_{\alpha}^{\epsilon}\,.

(7.23)

Finally, by summing over $\mathcal{C}_{\alpha}$ we get the estimate

\displaystyle\int_{B_{1}(p)}|\nabla^{2}u|^{q}\leq C(n,{\rm v},p)\lambda^{q}\,\sum_{\alpha}r_{\alpha}^{\epsilon}=C\,\sum 2^{-\epsilon\alpha}=C(n,{\rm v},q)\,,

(7.24)

as claimed. ∎

[01Z3]

8. Improved Estimates in Dimension 4

In this section we apply the codimension $4$ estimates of Theorem 1.1 in order to prove the finite diffeomorphism and $L^{2}$ curvature bounds of Theorem 1.5 and Theorem 1.4.

In subsection 8.1, we recall some necessary some preliminaries.

In subsection 8.2, we use the codimension $4$ estimate of Theorem 1.1 to prove the existence of good annuli which have curvature and harmonic radius control.

In subsection 8.3 we first use this to show that in the noncollapsed situation, at any point we have that away from a definite number of scales, every annulus is good. We combine this with a counting argument, which plays the role of an effective version of the fact any infinite collection of points has a limit point, in order to prove the harmonic radius estimates of Theorem 1.3.

In subsection 8.4 we prove the finite diffeomorphism statement of Theorem 1.4. Morally, the argument is quite similar to the one in [AnCh], though it is designed to be more effective in nature. In fact, the argument in Section 8.4 is quite general and works for any collection of uniformly noncollapsed smooth manifolds with bounded Ricci curvature, such that all Gromov-Hausdorff limits and blow ups only isolated singularities.

In subsection 8.5, we give a local version of the finite diffeomorphism theorem. Our main application of this is to prove a priori $L^{2}$ estimates on the curvature on a noncollapsed $4$ -manifold with bounded Ricci curvature.

[01Z4]

8.1. Diffeomorphisms and Harmonic Radius

To control the diffeomorphism type of a manifold, or of part of a manifold, the basic tool one needs is to control the total number of coordinate charts, the number of domains these charts which can intersect a given chart and the change of coordinate maps between these charts in a suitably strong topology. This type of result has a long history, going back to [Ch1] in the context of bounded sectional curvature. In particular, control on the harmonic radius enables one to implement such an argument.

In this subsection we recall two theorems that will be used later. We refer the reader to the book [P] for proofs of these statements. The first theorem states that when two manifolds with harmonic radius bounded from below are sufficiently Gromov-Hausdorff close, then they must be diffeomorphic.

[01Z5]

Theorem 8.1.

For every $\epsilon>0$ , there exists $\delta=\delta(n,\epsilon)$ , such that the following holds. If $M^{n}_{1},M^{n}_{2}$ are Riemannian manifolds and $U_{j}\subset M_{j}$ are subsets such that $r_{h}(x)>r>0$ for each $x\in U_{j}$ , and

d_{GH}(B_{r}(U_{1}),B_{r}(U_{2}))<\epsilon r\,,

then there exist open sets $B_{r/2}(U_{j})\subseteq U^{\prime}_{j}\subseteq B_{r}(U_{j})$ and a $C^{2}$ diffeomorphism $\Phi:U^{\prime}_{1}\to U^{\prime}_{2}$ , such that

\displaystyle||g_{1}-\Phi^{*}g_{2}||_{C^{0}}<\epsilon\,.

(8.1)

If we further assume $|{\rm Ric}_{M^{n}_{j}}|\leq n-1$ , $j=1,2$ , then $\Phi$ is in $C^{2,\alpha}\cap W^{3,q}$ for all $\alpha<1$ and $q<\infty$ , and in harmonic coordinates on $U^{\prime}_{1}$ we have

\displaystyle||g_{1}-\Phi^{*}g_{2}||_{C^{0}}+r^{1+\alpha}||\partial_{i}\Phi^{*}g_{2}||_{C^{\alpha}}+r^{2}||\partial_{i}\partial_{j}\Phi^{*}g_{2}||_{L^{q}}\leq C(n,\alpha,q)\epsilon\,.

(8.2)

The idea of the proof of Theorem 8.1 is to cover the set $U_{1}$ by harmonic charts $B_{r/2}(x_{j})$ of definite size, the intersection of whose domains also have a definite size or are empty and such that each chart domain intersects at most a definite number of distinct chart domains. By restricting the Gromov-Hausdorff map $f:U_{1}\to U_{2}$ to $U_{1}$ , and using that the image of each ball $f(B_{r}(x_{j}))$ lies in a harmonic coordinate chart of $U_{2}$ , we can construct a suitable smooth approximation of $f$ . Then using the estimates of the local charts one can see this smoothing of $f$ is the required diffeomorphism.

In a related direction, instead of trying to use the harmonic radius to directly to construct diffeomorphisms between nearby manifolds, we can use it to simply bound the number of diffeomorphism types of a space. Precisely, we have the following:

[01Z6]

Theorem 8.2.

There exists $C=C(n,D)$ with the following property. Let $(M^{n},g)$ denote a Riemannian manifold and $U\subseteq M$ a subset such that $r_{h}(x)>r>0$ for all $x\in U$ and such that ${\rm diam}(U)\leq D\cdot r$ . Then there exists an open set $U^{\prime}$ with $T_{r/2}(U)\subseteq U^{\prime}\subseteq T_{r}(U)$ , such that $U^{\prime}$ has at most one of $C$ diffeomorphism types.

The idea of the proof of the above is that $U$ may be covered by a controlled number of harmonic charts $B_{r/2}(x_{j})$ with suitable control as above on the intersections of their domains. The geometry estimates on the charts automatically imply control over the transition functions between these charts. Hence there are a finite number of ways this finite collection of balls can be pasted together.

[01Z7]

8.2. Annulus Estimates

In this section, we use Theorem 1.1 in order to prove our basic annulus estimates on $4$ -manifolds with bounded Ricci curvature. These are the key first steps toward the finite diffeomorphism statements and the corresponding curvature estimates of Theorem 1.5. To state our main result for this subsection let us recall the volume ratio

\displaystyle\mathcal{V}^{\delta}_{r}(x):=-\ln\left(\frac{{\rm Vol}(B_{r}(x))}{{\rm Vol}(B_{r}(0^{4}_{-\delta}))}\right)\,,

(8.3)

where $0^{4}_{-\delta}$ is a base point in the $4$ -dimensional hyperbolic space of constant curvature $-\delta$ ; by the Bishop-Gromov theorem, this ratio is monotone increasing for a manifold with Ricci curvature bounded from below ${\rm Ric}_{M^{n}}\geq-3\delta$ . It has been understood since [ChCo1] that almost constancy of $\mathcal{V}^{\delta}_{r}(x)$ over a range of scales leads to cone behavior of the underlying metric space. Our main result of this subsection states that in the context of bounded Ricci curvature and dimension $4$ , almost constancy of this volume ratio leads to much stronger control up to diffeomorphism and pointwise geometric control.

[01Z8]

Theorem 8.3.

For every $\epsilon>0$ there exists $\delta({\rm v},\epsilon)>0$ such that if $M^{4}$ satisfies $|{\rm Ric}_{M^{4}}|\leq 3\delta$ , ${\rm Vol}(B_{1}(p))>{\rm v}>0$ and $|\mathcal{V}^{\delta}_{4}(p)-\mathcal{V}^{\delta}_{1/4}(p)|<\delta$ , then there exists a discrete subgroup $\Gamma\subseteq{\rm O}(4)$ with $|\Gamma|\leq N({\rm v})$ such that the following hold:

(1)

For each $x\in A_{\epsilon,2}(p)$ we have the harmonic radius lower bound $r_{h}(x)>r_{0}({\rm v})\epsilon$ .
(2)

There exists a subset $A_{\epsilon,2}(p)\subseteq U\subseteq A_{\epsilon/2,2+\epsilon}(p)$ and a diffeomorphism $\Phi:A_{\epsilon,2}(0)\to U$ , with $0\in\mathds{R}^{4}/\Gamma$ , such that if $g_{ij}=\Phi^{*}g$ is the pullback metric then

$\displaystyle||g_{ij}-\delta_{ij}||_{C^{0}}+||\partial_{k}g_{ij}||_{C^{0}}<\epsilon\,.$ (8.4)

[01Z9]

Proof.

The proof is by contradiction. So let us assume for some $\epsilon>0$ there is no such $\delta({\rm v},\epsilon)>0$ . Thus, we have a sequence of spaces $(M^{4}_{j},g_{j},p_{j})$ with ${\rm Vol}(B_{1}(p_{j}))>{\rm v}>0$ , $|{\rm Ric}_{M^{4}_{j}}|\leq\delta_{j}\to 0$ and $|\mathcal{V}_{4}(p_{j})-\mathcal{V}_{1/4}(p_{j})|<\delta_{j}\to 0$ , but the conclusions of the theorem fail. After passing to a subsequence we can take a limit

\displaystyle(M^{4}_{j},d_{j},p_{j})\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}(X,d,p)\,.

(8.5)

Using the almost volume cone implies almost metric cone theorem of [ChCo1], we then have

\displaystyle B_{4}(p)=B_{4}\big(y_{0})\,,

(8.6)

where $y_{0}\in C(Y)$ is the cone vertex and $Y$ some metric space of diameter $\leq\pi$ .

Now using Theorem 1.1, we know that away from a set of codimension $4$ in $C(Y)$ , the harmonic radius $r_{h}>0$ is bounded uniformly from below. Assume there is some point $y\in Y$ such that $r_{h}(y)=0$ and consider the ray $\gamma_{y}$ in $C(Y)$ through the point $y$ . In that case, it would follow that for every point of $\gamma_{y}$ , the harmonic radius $r_{h}=0$ vanishes. The ray $\gamma$ has Hausdorff dimension $1$ , and therefore its existence would contradict Theorem 1.1. Thus, we conclude that $r_{h}>0$ and that $Y=(Y,g_{Y})$ is a $C^{1,\alpha}\cap W^{2,q}$ manifold for every $\alpha<1$ and $q<\infty$ .

Now by writing the formula for the Ricci tensor in harmonic coordinates and using $|{\rm Ric}_{M^{4}_{j}}|\to 0$ , it follows that $C(Y)$ is smooth and Ricci flat away from the vertex. In particular, since $C(Y)$ is a metric cone over $Y$ , we must ${\rm Ric}_{Y^{3}}=3g^{Y}$ . Since in dimension $3$ , constant Ricci curvature implies constant sectional curvature, it follows $Y=S^{3}/\Gamma$ has constant sectional curvature $\equiv 1$ . Additionally, we know from the volume bound, ${\rm Vol}(B_{1}(p))>{\rm v}>0$ , that the order $|\Gamma|<N({\rm v})$ is uniformly bounded. In particular, we have that $C(Y)=\mathds{R}^{4}/\Gamma$ is an orbifold with an isolated singularity.

It now follows that there exists $r_{0}({\rm v})>0$ such that for $y\in\mathds{R}^{4}/\Gamma$ with $|y|=1$ , we have

\displaystyle B_{2r_{0}}(y)=B_{2r_{0}}(0^{4})\,,

(8.7)

where $0^{4}\in\mathds{R}^{4}$ . In particular, for all $j$ sufficiently large, we have from the standard $\epsilon$ -regularity theorem, Theorem 2.3, that for all $x\in A_{\epsilon,2}(p_{j})$ , the harmonic radius, $r_{h}(x)>r_{0}({\rm v},\epsilon)=r_{0}({\rm v})\epsilon$ is bounded uniformly from below independent of $j$ . Thus, if there exists $\epsilon$ as above, for which there is no $\delta({\rm v},\epsilon)$ , it must be (2) that fails to hold.

However, by using again the diffeomorphism statement of Theorem 8.1, we have that for $j$ sufficiently large, there exists diffeomorphisms

\displaystyle\Phi_{j}:A_{\epsilon,2}(0)\to M^{4}_{j}\,,

(8.8)

such that

\displaystyle\Phi_{j}^{*}g_{j}\stackrel{{\scriptstyle C^{1,\alpha}\cap W^{2,q}}}{{\longrightarrow}}dr^{2}+r^{2}g_{Y}\,.

(8.9)

For $j$ sufficiently large, this implies that (2) holds; a contradiction. ∎

[01ZA]

8.3. Regularity Scale Estimates

In this subsection we prove the harmonic and regularity scale estimates (1.9) of Theorem 1.5. We know already from Theorem 1.1 that if $M^{4}\to X$ is a limit space, then the singular set of $X$ has dimension zero. The estimate (1.9) may be viewed as an effective version of this statement. Indeed, (1.9) not only gives a bound on the number of singularities which can appear, but it gives a bound on the number of balls with large curvature concentration. Motivated by Theorem 8.3 and the constructions of [ChNa13], we begin with the following definition which will be useful in subsequent sections as well.

[01ZB]

Definition 8.4.

Consider the scales $r_{\alpha}=2^{-\alpha}$ . For each $x\in M$ we associate the infinite tuple $T(x)\in\mathds{Z}_{2}^{\mathds{N}}$ defined by

\displaystyle T_{\alpha}(x)\equiv\begin{cases}1\text{ if }|\mathcal{V}^{\delta}_{4r_{\alpha}}(x)-\mathcal{V}^{\delta}_{r_{\alpha}/4}(x)|\geq\delta\,\\ 0\text{ if }|\mathcal{V}^{\delta}_{4r_{\alpha}}(x)-\mathcal{V}^{\delta}_{r_{\alpha}/4}(x)|<\delta\,.\end{cases}

We denote by $|T|(x)=\sum T_{\alpha}(x)$ the number of bad scales at $x\in M^{4}$ .

[01ZC]

Remark 8.1.

The definition of $T(x)$ relies on a choice of $\delta>0$ . When we want to stress this, we will write $T^{\delta}(x)$ , but otherwise will supress this dependence.

We begin with the following; see also [ChNa13] for the same statement in a more general context:

[01ZD]

Lemma 8.5.

Let ${\rm Ric}_{M^{4}}\geq-3\delta$ and ${\rm Vol}(B_{1}(p))>{\rm v}>0$ with $\delta\leq 1$ . Then for each $\delta^{\prime}>0$ and $x\in B_{2}(p)$ there exists at most $N({\rm v},\delta^{\prime})$ scales $\alpha\in\mathds{N}$ such that

\displaystyle\big|\mathcal{V}^{\delta}_{r_{\alpha+1}}(x)-\mathcal{V}^{\delta}_{r_{\alpha}}(x)\big|<\delta^{\prime}\,.

(8.10)

[01ZE]

Proof.

For $x\in B_{2}(p)$ fixed, we have

\displaystyle{\rm Vol}(B_{1}(x))\geq C(n)^{-1}{\rm Vol}(B_{3}(x))\geq C^{-1}{\rm Vol}(B_{1}(p))\geq C^{-1}{\rm v}>0\,,

(8.11)

and so,

\displaystyle\mathcal{V}^{\delta}_{1}(x)\leq-\ln\Big(C^{-1}{\rm v}\Big)=C(n,{\rm v})\,.

(8.12)

From the monotonicity of $\mathcal{V}^{\delta}_{r}(x)$ , we have

	$\displaystyle C(n,{\rm v})-1\geq\mathcal{V}^{\delta}_{1}(x)-\mathcal{V}^{\delta}_{0}(x)$	$\displaystyle=\sum\Big(\mathcal{V}^{\delta}_{r_{\alpha}}(x)-\mathcal{V}^{\delta}_{r_{\alpha+1}}(x)\Big)$
		$\displaystyle=\sum\Big\|\mathcal{V}^{\delta}_{r_{\alpha}}(x)-\mathcal{V}^{\delta}_{r_{\alpha+1}}(x)\Big\|\,.$		(8.13)

In particular, there are at most $N=C(n,{\rm v})(\delta^{\prime})^{-1}$ elements $\alpha\in\mathds{N}$ such that

\displaystyle\Big|\mathcal{V}_{r_{\alpha}}(x)-\mathcal{V}_{r_{\alpha+1}}(x)\Big|>\delta^{\prime}\,,

(8.14)

as claimed. ∎

Let us point out the following useful corollary:

[01ZF]

Corollary 8.6.

Let $M^{4}$ satisfy ${\rm Ric}_{M^{4}}\geq-3\delta$ and ${\rm Vol}(B_{1}(p))>{\rm v}>0$ . Then for each $x\in B_{2}(p)$ we have

\displaystyle|T^{\delta}|(x)\leq N({\rm v},\delta)\,.

(8.15)

[01ZG]

Proof.

Put $\delta^{\prime}=\delta/3$ . Then for $x\in B_{2}(p)$ , there are at most $N({\rm v},\delta)=C({\rm v})\delta^{-1}$ scales $\alpha$ for which

\displaystyle\Big|\mathcal{V}_{r_{\alpha}}(x)-\mathcal{V}_{r_{\alpha+1}}(x)\Big|>\frac{\delta}{3}\,.

(8.16)

Hence, there are at most $3N$ elements $\alpha\in\mathds{N}$ such that

\displaystyle\Big|\mathcal{V}_{r_{\beta}}(x)-\mathcal{V}_{r_{\beta+1}}(x)\Big|>\frac{\delta}{3}\,,

(8.17)

for some $\beta\in\{\alpha-1,\alpha,\alpha+1\}$ . Therefore, for all other $\alpha$ , we must have

\displaystyle\Big|\mathcal{V}_{4r_{\alpha}}(x)-\mathcal{V}_{r_{\alpha}/4}(x)\Big|<\delta\,,

(8.18)

which proves the corollary. ∎

We end this subsection with a proof of the regularity scale estimate (1.9) from Theorem 1.5. One can view the proof as an effective version of the fact that an infinite collection of points must have a limit point.

[01ZH]

Proof of Estimate (1.9) of Theorem 1.5.

Let $M^{n}$ satisfy $|{\rm Ric}_{M^{4}}|\leq 3$ and ${\rm Vol}(B_{1}(p))>{\rm v}>0$ . We will prove the estimate for the harmonic radius $r_{h}$ . The same argument works in the Einstein case to control the regularity scale.

So let $0<\epsilon<<1$ be fixed with $\delta(n,\epsilon)$ chosen to satisfy Theorem 8.3. Consider the set

\displaystyle\{x\in B_{1}(p):r_{h}(x)<r\}\,.

(8.19)

In view of the doubling condition implied by the Bishop-Gromov inequality, we have by a standard construction that there exists a covering $\{B_{r}(x_{j})\}_{1}^{N}$ with

\displaystyle\{x\in B_{1}(p):r_{h}(x)<r\}\subseteq\bigcup_{j}B_{r}(x_{j})\,,

(8.20)

but such that $\{B_{r/4}(x_{j})\}$ are disjoint. Such coverings, which we will term “efficient”, will be constructed several times below. Note that

\displaystyle T_{r}\big(\{x\in B_{1}(p):r_{h}(x)<r\}\big)\subseteq\bigcup_{j}B_{2r}(x_{j})\,,

(8.21)

and thus

\displaystyle{\rm Vol}\Big(T_{r}\big(\{x\in B_{1}(p):r_{h}(x)<r\}\big)\Big)\leq\sum_{1}^{N}{\rm Vol}(B_{2r}(x_{j}))\leq C(n)N\cdot r^{4}\,.

(8.22)

Hence, our goal is to control the number of balls $N$ in the covering. Denote by $\mathcal{C}\equiv\{x_{j}\}_{1}^{N}$ this collection of points.

Now note the following: if $x_{j}$ is one of our ball centers and $T_{\alpha}(x_{j})=0$ , then by Theorem 8.3 we have for every $x\in A_{r_{\alpha}/2,2r_{\alpha}}(x_{j})$ that $r_{h}(x)>\bar{r}({\rm v})\cdot r_{\alpha}$ . In particular, if $r_{\alpha}>\bar{r}^{-1}r$ , this implies that

\displaystyle x_{k}\not\in A_{r_{\alpha}/2,2r_{\alpha}}(x)\,.

(8.23)

Now let us inductively build a sequence of decreasing subsets $\mathcal{C}^{k+1}\subseteq\mathcal{C}^{k}\subseteq\cdots\subseteq\mathcal{C}$ and associated radii $s_{k}=r_{\alpha_{k}}>0$ with ${\rm diam}(\mathcal{C}^{k})<4s_{k}$ . There are three key inductive properties that will be proved about these sets:

(1)

There exists $C(n)>0$ such that the cardinality of $\mathcal{C}^{k}$ satisfies

$\displaystyle\big|\#\mathcal{C}^{k}\big|\geq C^{-k}\big|\#\mathcal{C}\big|=C^{-k}N\,.$ (8.24)
(2)

For every $x^{k}_{j}\in\mathcal{C}^{k}$ we have

$\displaystyle\sum_{0\leq\alpha\leq\alpha_{k}}T^{\delta}_{\alpha}(x^{k}_{j})\geq k\,.$ (8.25)
(3)

If $\big|\#\mathcal{C}^{k}\big|>1$ and $s_{k}>\bar{r}^{-1}r$ then $\mathcal{C}^{k+1}\neq\emptyset$ .

Before constructing the sequence of sets, let us see that once the construction is complete, we will have proved our desired estimate on $N$ . Indeed, let $k$ be the largest index such that $\mathcal{C}^{k}\neq\emptyset$ . By the third property we must have either $|\#\mathcal{C}^{k}|=1$ or $s_{k}\leq\bar{r}^{-1}r$ , at which point we get by a covering argument that $|\#\mathcal{C}^{k}|<C(n)$ . By Lemma 8.5 and the second property we have that $k\leq k(n,{\rm v},\delta)=k(n,{\rm v})$ , and thus by the first property we have

\displaystyle N\leq C(n)^{k(n,{\rm v})}\cdot|\#\mathcal{C}^{k}|\leq C(n,{\rm v})\,,

(8.26)

which proves the result.

Now let $\mathcal{C}^{0}\equiv\mathcal{C}$ with $s_{0}=1$ . Clearly, the inductive properties hold for $\mathcal{C}^{0}$ . Assume we have built $\mathcal{C}^{k}\subseteq\mathcal{C}$ with $s_{k}>0$ satisfying the inductive properties, and let us build $\mathcal{C}^{k+1}$ . First note that if $|\#\mathcal{C}^{k}|=1$ or $s_{k}\leq\bar{r}^{-1}r$ , then we let $\mathcal{C}^{k+1}=\emptyset$ . Our construction will otherwise give us a nonempty $\mathcal{C}^{k+1}$ , so that the third inductive property will automatically be satisfied. So let us denote $s^{\prime}_{k}={\rm diam}(\mathcal{C}^{k})\cdot 2^{-10}$ . Choose an efficient covering $\{B_{s^{\prime}_{k}}(x^{k}_{j})\}$ , where $x^{k}_{j}\in S^{k}$ , so that the balls in $\{B_{s^{\prime}_{k}/4}(x^{k}_{j})\}$ are disjoint. Note that because ${\rm diam}(\mathcal{C}^{k})<4s_{k}$ , the usual doubling estimates imply that there are at most $C(n)$ balls in this covering. We choose the ball $B_{s^{\prime}_{k}}(y)$ such that $\mathcal{C}^{k}\cap B_{s^{\prime}_{k}}(y)$ has the largest cardinality of any ball from the covering. Then we define $\mathcal{C}^{k+1}=\mathcal{C}^{k}\cap B_{s^{\prime}_{k}}(y)$ .

By that by our choice of ball, $B_{s^{\prime}_{k}}(y)$ , we have

\displaystyle\big|\#\mathcal{C}^{k+1}\big|

\displaystyle=\big|\#\mathcal{C}^{k}\cap B_{s^{\prime}_{k}}(y)\big|\geq C(n)^{-1}\big|\#\mathcal{C}^{k}\big|\geq C^{-(k+1)}N\,,

(8.27)

so that $\mathcal{C}^{k+1}$ satisfies the first inductive property. To find $s_{k+1}$ and prove the second inductive property, let us define the following. For each $x^{k+1}_{j}\in\mathcal{C}^{k+1}$ if

\displaystyle T^{\delta}_{\alpha_{k}+7}(x^{k+1}_{j})=1\,,

(8.28)

then let us set $\beta_{j}=\alpha_{k}+7$ , and otherwise let $\beta_{j}\geq\alpha_{k}+8$ be the largest integer such that $T^{\delta}_{\beta_{j}-1}(x^{k+1}_{j})=0$ but $T^{\delta}_{\beta_{j}}(x^{k+1}_{j})=1$ . Note that $B_{s^{\prime}_{k}}(y)\subseteq B_{r_{\alpha_{k}+7}}(x^{k}_{j})$ . Let $\alpha_{k+1}\equiv\max\{\beta_{j},\lceil-\log_{2}\big(\bar{r}r^{-1}\big)\rceil\}$ with $x^{k+1}\in\mathcal{C}^{k+1}$ the associated element which attains the maximum, and note by (8.23) that

\displaystyle\mathcal{C}^{k+1}=\mathcal{C}^{k}\cap B_{s^{\prime}_{k}}(y)=\mathcal{C}^{k}\cap B_{2^{-\alpha_{k+1}+1}}(x^{k+1})\,.

(8.29)

In particular, with $s_{k+1}=r_{\alpha_{k+1}}$ then ${\rm diam}(\mathcal{C}^{k+1})<4s_{k+1}$ , and the second inductive property holds, which completes the induction step of the construction, and hence, the proof.

∎

[01ZI]

8.4. Finite Diffeomorphism Type

In this subsection we will prove Theorem 1.4 and give some refinements which will be useful for the $L^{2}$ -curvature estimate of Theorem 1.5.

We begin by associating a good scale to the subgroup of ${\rm O}(4)$ occuring in Theorem 8.3.

[01ZJ]

Definition 8.7.

Let $\epsilon,\delta>0$ be such that Theorem 8.3 holds. For $x\in B_{1}(p)$ and $\alpha\in\mathds{N}$ such that $T^{\delta}_{\alpha}(x)=0$ , we denote by $\Gamma_{\alpha}(x)\subseteq O(4)$ , the uniquely defined discrete subgroup arising from Theorem 8.3.

The following is the key Neck lemma for our finite diffeomorphism of Theorem 1.4. In essence, the proof of Theorem 1.4 will come from decomposing $M$ into a finite number of distinct pieces. What we are refering to informally as the neck regions will be diffeomorphic to cylinders $\mathds{R}\times S^{3}/\Gamma$ . They will connect the pieces which will be refered to as body regions.

[01ZK]

Lemma 8.8.

For every $0<\epsilon\leq\epsilon({\rm v})$ , there exists $\delta=\delta({\rm v},\epsilon)$ with the following properties. Let $M^{4}$ satisfy $|{\rm Ric}_{M^{4}}|\leq 3\delta$ and ${\rm Vol}(B_{1}(p))>{\rm v}>0$ . Let $x\in B_{1}(p)$ and assume $\alpha_{1}\in\mathds{N}$ satisfies $T^{\delta}_{\alpha_{1}}(x)=0$ with $\Gamma_{\alpha_{1}}$ the corresponding group. Then if $\alpha_{2}\in\mathds{N}$ is such that $\mathcal{V}^{\delta}_{r_{\alpha_{2}}/4}(x)\geq\ln\big|\Gamma_{\alpha_{1}}\big|-\delta$ , there exists a subset $A_{r_{\alpha_{2}}/2,2r_{\alpha_{1}}}(x)\subseteq U\subseteq A_{(1-\epsilon)r_{\alpha_{2}}/2,2(1+\epsilon)r_{\alpha_{1}}}(x)$ and a diffeomorphism $\Phi:A_{r_{\alpha_{2}}/2,2r_{\alpha_{1}}}(0)\to U$ , where $0\in\mathds{R}^{4}/\Gamma_{\alpha_{1}}$ , such that if $g_{ij}=\Phi^{*}g$ is the pullback metric, we have

\displaystyle||g_{ij}-\delta_{ij}||_{C^{0}(A_{r_{\alpha}/2},r_{2\alpha})}+r_{\alpha}||\partial_{k}g_{ij}||_{C^{0}(A_{r_{\alpha}/2},2r_{\alpha})}<\epsilon

(8.30)

[01ZL]

Proof.

We will fix $\epsilon({\rm v})>0$ later. For the moment let any $\epsilon_{1}>0$ be arbitrary with $\delta_{1}({\rm v},\epsilon_{1})>0$ the corresponding number from Theorem 8.3. If $T^{\delta_{1}}_{\alpha_{1}}(x)=0$ then there exists a diffeomorphism

\displaystyle\Phi_{\alpha_{1}}:A_{r_{\alpha_{1}}/2,2r_{\alpha_{1}}}(0)\to U_{\alpha_{1}}\,,

(8.31)

where $0\in\mathds{R}^{4}/\Gamma_{\alpha_{1}}$ and $A_{r_{\alpha_{1}}/2,2r_{\alpha_{1}}}(x)\subseteq U_{\alpha}\subseteq A_{(1-\epsilon)r_{\alpha_{1}}/2,(1+\epsilon)r_{\alpha_{1}}}(x)$ , such that

\displaystyle||\Phi_{\alpha_{1}}^{*}g_{ij}-\delta_{ij}||_{C^{0}(A_{r_{\alpha_{1}}/2},2r_{\alpha_{1}})}+r_{\alpha_{1}}||\partial_{k}\Phi_{\alpha_{1}}^{*}g_{ij}||_{C^{0}(A_{r_{\alpha_{1}}/2},2r_{\alpha_{1}})}<\epsilon_{1}\,.

(8.32)

In particular, if $\epsilon>0$ is fixed and $2\delta(n,\epsilon)$ is the corresponding number from Theorem 8.3, then we can choose $\epsilon_{1}=\epsilon_{1}(\epsilon,{\rm v})$ sufficiently small so that

\displaystyle\mathcal{V}^{\delta}_{r_{\alpha_{1}}}(x)<\ln|\Gamma_{\alpha_{1}}|+\delta\,.

(8.33)

Thus, if $\alpha_{2}$ is such that

\displaystyle\mathcal{V}^{\delta}_{r_{\alpha_{2}}/2}(x)\geq\ln|\Gamma_{\alpha_{1}}|-\delta\,,

(8.34)

then for all $\alpha_{1}\leq\alpha\leq\alpha_{2}$ we have $T^{2\delta}_{\alpha}(x)=0$ .

By Theorem 8.3, there exists for each $\alpha_{1}\leq\alpha\leq\alpha_{2}$ , a diffeomorphism

\displaystyle\Phi_{\alpha}:A_{r_{\alpha}/2,2r_{\alpha}}(0)\to U_{\alpha}\,,

(8.35)

where $0\in\mathds{R}^{4}/\Gamma_{\alpha}$ and $A_{r_{\alpha}/2,2r_{\alpha}}(x)\subseteq U_{\alpha}\subseteq A_{(1-\epsilon)r_{\alpha}/2,2(1+\epsilon)r_{\alpha}}(x)$ , such that

\displaystyle||\Phi_{\alpha}^{*}g_{ij}-\delta_{ij}||_{C^{0}(A_{r_{\alpha}/2},2r_{\alpha})}+r_{\alpha}||\partial_{k}\Phi_{\alpha}^{*}g_{ij}||_{C^{0}(A_{r_{\alpha}/2},2r_{\alpha})}<\epsilon\,.

(8.36)

In particular this implies that $\Gamma_{\alpha}=\Gamma$ is independent of $\alpha$ .

Next we focus on the inverse maps

\displaystyle\Phi_{\alpha}^{-1}:U_{\alpha}\to A_{r_{\alpha}/2,2r_{\alpha}}(0)\,.

(8.37)

Observe that by (8.36), after possibly composing $\Phi_{\alpha}$ with a rotation of $\mathds{R}^{4}/\Gamma$ we can assume for $x\in U_{\alpha}\cap U_{\beta}$ that

\displaystyle|\Phi_{\alpha}^{-1}(x)-\Phi_{\beta}^{-1}(x)|<\epsilon r_{\alpha}\,.

(8.38)

Now let $\epsilon<\epsilon({\rm v})$ be sufficiently small, so that if $x\in\mathds{R}^{4}/\Gamma$ , then $B_{\epsilon|x|}(x)\subseteq\mathds{R}^{4}/\Gamma$ is isometric to the standard Euclidean ball $B_{\epsilon|x|}(0^{4})\subseteq\mathds{R}^{4}$ . Note in particular that if $\{x_{i}\}\in B_{\epsilon|x|}(x)$ is a collection of points, then any convex combination is well defined.

For each $\alpha$ let $\varphi^{\prime}_{\alpha}:U_{\alpha}\to\mathds{R}$ be a smooth cutoff function such that

\displaystyle\varphi^{\prime}_{\alpha}(x)=\begin{cases}&1\text{ if }x\in A_{3r_{\alpha}/8,15r_{\alpha}/8}(x)\,,\\ &0\text{ if }x\not\in A_{r_{\alpha}/2,2r_{\alpha}}(x)\,,\end{cases}

and such that $|\nabla\varphi^{\prime}_{\alpha}|\leq 10r_{\alpha}^{-1}$ . If we set $\varphi^{\prime}(x)=\sum_{\alpha}\varphi^{\prime}_{\alpha}(x)$ then $1\leq\varphi^{\prime}(x)\leq 4$ . In particular,

\displaystyle\varphi_{\alpha}=\frac{\varphi^{\prime}_{\alpha}(x)}{\varphi^{\prime}(x)}:U_{\alpha}\to\mathds{R}\,,

(8.39)

sarisfies $\sum\varphi_{\alpha}(x)=1$ , and so, is a partition of unity, with $|\nabla\varphi_{\alpha}|\leq 40r_{\alpha}^{-1}$ .

Define the map

\displaystyle\Phi^{-1}:U=\bigcup_{\alpha}U_{\alpha}\to A_{r_{\alpha_{2}}/2,2r_{\alpha_{1}}}(0)\,,

(8.40)

given by

\displaystyle\Phi^{-1}(x)=\sum_{\alpha}\varphi_{\alpha}(x)\Phi^{-1}_{\alpha}(x)\,.

(8.41)

(As previously noted, the convex combination is well defined since the $\Phi^{-1}_{\alpha}(x)$ all live in a ball which is isometric to a Euclidean ball.) On each domain, $U_{\alpha}$ , we have by (8.32) and (8.38) that $\Phi^{-1}$ and $\Phi^{-1}_{\alpha}$ are $C^{1}$ -close. Hence, $\Phi^{-1}$ is a diffeomorphism, and a quick computation using (8.32) and (8.38) verifies the desired estimates:

\displaystyle||\Phi^{*}g_{ij}-\delta_{ij}||_{C^{0}(A_{r_{\alpha}/2},2r_{\alpha})}+r_{\alpha}||\partial_{k}\Phi_{\alpha}^{*}g_{ij}||_{C^{0}(A_{r_{\alpha}/2},2r_{\alpha})}<C\epsilon\,.

(8.42)

By choosing $\epsilon$ appropriately small, we complete the proof. ∎

The following lemma could be termed a “gap lemma”. It will be used to tell us that if we consider two distinct neck regions, then the complexity of the smaller neck region must be strictly less than that of the larger neck region.

[01ZM]

Lemma 8.9.

For each $\delta<\delta({\rm v})$ , there exists $\bar{\alpha}(\delta,{\rm v})$ with the following property. If $|{\rm Ric}_{M^{4}}|\leq 3\delta$ , ${\rm Vol}(B_{1}(p))>{\rm v}>0$ , and $\mathcal{V}^{\delta}_{r_{1}}(x)<\ln N-\delta$ for some $N\in\mathds{N}$ and $x\in B_{1}(p)$ , we have

\mathcal{V}^{\delta}_{r_{\bar{\alpha}}}(x)<\ln\Big(N-1\Big)+\delta\,.

[01ZN]

Proof.

First note by Theorem 8.3 that if $\delta$ is fixed, then there exists $\delta^{\prime}({\rm v},\delta)$ such that if $|{\rm Ric}|\leq 3\delta^{\prime}$ and if $T^{\delta^{\prime}}_{0}=0$ , then

\displaystyle\big|\mathcal{V}^{\delta^{\prime}}_{1}(x)-\ln|\Gamma_{0}|\big|<\delta\,.

(8.43)

By rescaling this inequality, we see that in the context of this lemma, the following holds. If $x\in B_{1}(p)$ , $\alpha>\bar{\alpha}(v,\delta)$ and

\displaystyle\big|\mathcal{V}^{\delta}_{4r_{\alpha}}(x)-\mathcal{V}^{\delta}_{r_{\alpha}/4}(x)\big|<\delta^{\prime}\,,

(8.44)

then we have

\displaystyle\big|\mathcal{V}^{\delta}_{r_{\alpha}}(x)-\ln|\Gamma_{\alpha}|\big|<\delta\,.

(8.45)

In particular, for $x\in B_{1}(p)$ , we can apply Lemma 8.5 to see that there exists a scale $\alpha\leq\bar{\alpha}(v,\delta)$ such that

\displaystyle\big|V^{\delta}_{r_{\alpha}}(x)-\mathcal{V}^{\delta}_{r_{\alpha+1}}(x)\big|<\delta^{\prime}\,,

(8.46)

and hence

\displaystyle\big|\mathcal{V}^{\delta}_{r_{\alpha}}(x)-\ln|\Gamma_{\alpha}|\big|<\delta\,.

(8.47)

However, if

\displaystyle\mathcal{V}^{\delta}_{r_{\alpha}}(x)<\ln N-\delta\,,

(8.48)

this implies $|\Gamma_{\alpha}|<N$ , which completes the proof. ∎

In Lemma 8.8 we have built the required structure for constructting the neck regions of our decomposition. What is left is to be able to build the body regions of the decomposition. The following lemma will be applied in the proof of Theorem 1.4 in order to construct the various body regions.

[01ZP]

Lemma 8.10.

For every $\delta>0$ , there exists $r_{0}({\rm v},\delta),N({\rm v},\delta)>0$ with the following properties. Let $M^{4}$ satisfy $|{\rm Ric}_{M^{4}_{j}}|\leq 3\delta$ , ${\rm Vol}(B_{1}(p))>{\rm v}>0$ . Then there exists points $\{x_{j}\}_{1}^{N}$ with $N\leq N({\rm v},\delta)$ , and scales $\alpha_{j}\in\mathds{N}$ with $r_{j}\equiv r_{\alpha_{j}}>r_{0}$ , such that

(1)

$T^{\delta}_{\alpha_{j}}(x_{j})=0$ ,
(2)

If $x\in B_{1}(p)\setminus\bigcup_{j}B_{r_{j}}(x_{j})$ then $r_{h}(x)>r_{0}$ ,
(3)

If $\beta_{j}\in\mathds{N}$ denotes the largest integer such that $\mathcal{V}^{\delta}_{r_{\beta_{j}}/4}(x_{j})\geq\ln|\Gamma_{j}|-\delta$ , then for every $x\in B_{2r_{\beta_{j}}}(x_{j})$ we have

$\displaystyle\mathcal{V}^{\delta}_{r_{\beta_{j}}/8}(x)<\ln|\Gamma_{j}|-\delta\,.$ (8.49)

[01ZQ]

Proof.

Let $\delta>0$ be chosen with $\delta^{\prime}({\rm v},\delta)$ to be chosen later. Note that by Lemma 8.5, for each $x\in B_{1}(p)$ there exists $\alpha_{x}\leq\bar{\alpha}(v,\delta^{\prime})$ such that $T^{\delta^{\prime}}_{\alpha_{x}}(x)=0$ . Consider the covering $\{B_{r_{\alpha_{x}}}(x)\}$ of $B_{1}(p)$ , and choose an efficient subcovering $\{B_{r_{j}}(x^{\prime}_{j})\}_{1}^{N}$ , where $r_{j}=r_{\alpha_{x^{\prime}_{j}}}$ and the balls in $\{B_{r_{j}/4}(x^{\prime}_{j})\}$ are disjoint. The usual doubling arguments imply that $N\leq N(v,\delta^{\prime})$ .

By Theorem 8.3, if we are given $\epsilon>0$ , then we can choose $\delta^{\prime}({\rm v},\epsilon,\delta)$ such that for each $x\in B_{\epsilon r_{j}}(x^{\prime}_{j})$ we have $T^{\delta}_{\alpha_{j}}(x)=0$ , while for each $x\in A_{\epsilon r_{j},2r_{j}}(x_{j})$ we have $r_{h}(x)>\bar{r}(v,\epsilon)r_{j}\geq r_{0}(v,\epsilon,\delta^{\prime})$ . Let $\Gamma_{j}$ be the group associated to $B_{r_{j}}(x^{\prime}_{j})$ , and for each $x\in B_{\epsilon r_{j}}(x^{\prime}_{j})$ let $\beta_{j}(x)$ be the largest integer such that $V^{\delta}_{r_{\beta_{j}}/4}(x)\geq\ln|\Gamma_{j}|-\delta$ . Let $\beta_{j}=\max\beta_{j}(x)$ with $x_{j}$ the corresponding point. Note that for $\epsilon(v,\delta)$ sufficiently small, we have $B_{2r_{\beta_{j}}}(x_{j})\subseteq B_{\epsilon r_{j}}(x^{\prime}_{j})$ , and in particular, for every $x\in B_{2r_{\beta_{j}}}(x_{j})$

\displaystyle V^{\delta}_{r_{\beta_{j}}/8}(x)<\ln|\Gamma_{j}|-\delta\,.

(8.50)

Consider the collection of balls $\{B_{r_{j}}(x_{j})\}$ . Clearly, by construction, conditions (1) and (3) are satisfied. If $x\in B_{1}(p)\setminus\{B_{r_{j}}(x_{j})\}$ then since $\{B_{2r_{j}}(x_{j})\}$ cover $B_{1}(p)$ we have that for some $x_{j}$ that $x\in A_{r_{j},2r_{j}}(x_{j})$ , which implies $r_{h}(x)\geq r_{0}(v,\delta)$ , as claimed. ∎

By the previous lemma, the regions between necks, namely $B_{1}(p)\setminus\bigcup_{j}B_{r_{\alpha_{j}}}(x_{j})$ , can be written as the union of a definite number of balls of definite size, on which there is definite geometric control.

We are nearly in a position to prove Theorem 1.4. To do so we will in fact prove the following stronger result, which is the bubble tree decomposition of $M^{4}$ .

[01ZR]

Theorem 8.11.

Let $M^{4}$ satisfy $|{\rm Ric}_{M^{4}}|\leq 3$ , $Vol(M)\geq{\rm v}>0$ and ${\rm diam}(M)\leq D$ . Then there exists a decomposition of $M^{4}$

\displaystyle M^{4}=\mathcal{B}^{1}\cup\bigcup_{j_{2}=1}^{N_{2}}\mathcal{N}^{2}_{j_{2}}\cup\bigcup_{j_{2}=1}^{N_{2}}\mathcal{B}^{2}_{j_{2}}\cup\cdots\cup\bigcup_{j_{k}=1}^{N_{k}}\mathcal{N}^{k}_{j_{k}}\cup\bigcup_{j_{k}=1}^{N_{k}}\mathcal{B}^{k}_{j_{k}}\,,

(8.51)

into open sets which satisfy the following:

(1)

If $x\in\mathcal{B}^{\ell}_{j}$ then $r_{h}(x)>r_{0}(n,{\rm v},D)\cdot{\rm diam}(\mathcal{B}^{\ell}_{j})$ .
(2)

Each neck $\mathcal{N}^{\ell}_{j}$ is diffeomorphic to $\mathds{R}\times S^{3}/\Gamma^{\ell}_{j}$ for some $\Gamma^{\ell}_{j}<O(4)$ .
(3)

$\mathcal{N}^{\ell}_{j}\cap\mathcal{B}^{\ell}_{j}$ is diffeomorphic to $\mathds{R}\times S^{3}/\Gamma^{\ell}_{j}$ .
(4)

$\mathcal{B}^{\ell-1}_{j^{\prime}}\cap\mathcal{N}^{\ell}_{j}$ are either empty or diffeomorphic to $\mathds{R}\times S^{3}/\Gamma^{\ell}_{j}$ .
(5)

$N_{\ell}\leq N({\rm v},D)$ and $k\leq k({\rm v},D)$ .

[01ZS]

Proof.

Let us remark first, that if $p\in M^{n}$ , then by volume ratio monotonicity, we have for every $r\leq 1$ that

\displaystyle{\rm Vol}(B_{r}(p))\geq\frac{{\rm Vol}_{-1}(B_{r})}{{\rm Vol}_{-1}(B_{D})}{\rm Vol}(B_{D}(p))\geq C(n,D)^{-1}{\rm Vol}(M^{4})r^{n}\geq C^{-1}{\rm v}r^{n}=:{\rm v}^{\prime}r^{n}\,.

(8.52)

Let $\epsilon<\epsilon({\rm v}^{\prime})$ from Lemma 8.8 with $\delta({\rm v},D,\epsilon)$ sufficiently small to satisfy Theorem 8.3 and Lemmas 8.8, 8.9, 8.10. After rescaling, it is sufficient to consider a Riemannian manifold $(M^{4},g)$ with $|{\rm Ric}_{M^{4}}|\leq 3\delta$ , ${\rm diam}(M)\leq D\delta^{-2}=D^{\prime}$ and ${\rm Vol}(B_{1}(p))>{\rm v}^{\prime}>0$ for every $p\in M$ .

Let us begin by efficiently covering $M^{4}$ by balls $\{B_{1}(x^{0}_{j})\}$ such that the balls in $\{B_{1/4}(x^{0}_{j})\}$ are disjoint. By the usual doubling argument, there are at most $N(n,D,{\rm v})$ such balls. For each such ball, we apply Lemma 8.10 in order to produce a collection of balls $\{B_{r^{1}_{j}}(x^{1}_{j})\}_{1}^{N_{1}}$ such that $r^{1}_{j}=r_{\alpha^{1}_{j}}>\bar{r}({\rm v},D)$ , $N_{1}\leq N({\rm v}^{\prime},D^{\prime})$ , $T^{\delta}_{\alpha^{1}_{j}}(x^{1}_{j})=0$ , and such that if $x\in M^{4}\setminus\bigcup_{j}B_{r^{1}_{j}}(x^{1}_{j})$ then $r_{h}(x)>\bar{r}$ . Furthermore, if we denote by $\Gamma^{2}_{j}$ , the group associated to $B_{r^{1}_{j}}(x^{1}_{j})$ , then if $\beta^{1}_{j}$ is the largest integer such that $V^{\delta}_{r_{\beta^{1}_{j}}/2}(x^{1}_{j})\geq\ln|\Gamma_{j}^{2}|-\delta$ , then for all $x\in B_{2r_{\beta^{1}_{j}}}(x^{1}_{j})$ we have

\displaystyle\mathcal{V}^{\delta}_{r_{\beta^{1}_{j}}/4}(x^{1}_{j})<\ln|\Gamma_{j}^{2}|-\delta\,.

(8.53)

Define

\displaystyle\mathcal{B}^{1}=:M^{4}\setminus\bigcup B_{r_{j}}(x_{j})\,,

(8.54)

as the first body region. Then we can write

\displaystyle M^{4}=\mathcal{B}^{1}\bigcup B_{2r^{1}_{j}}(x^{1}_{j})\,,

(8.55)

where by using Theorem 8.3, we have that $B_{2r^{1}_{j}}(x^{1}_{j})\cap\mathcal{B}^{1}$ is diffeomorphic to $\mathds{R}\times S^{3}/\Gamma^{1}_{j}$ .

Now to prove the theorem, let us inductively build a decomposition of $M^{4}$

\displaystyle M^{4}=\mathcal{B}^{1}\cup\bigcup_{j_{2}=1}^{N_{2}}\mathcal{N}^{2}_{j_{2}}\bigcup_{j_{2}=1}^{N_{2}}\mathcal{B}^{2}_{j_{2}}\cup\cdots\cup\bigcup_{j_{k}=1}^{N_{k}}\mathcal{N}^{k}_{j_{k}}\bigcup_{j_{k}=1}^{N_{k}}\mathcal{B}^{k}_{j_{k}}\cup\bigcup_{a=1}^{N_{k+1}}B_{2r^{k}_{a}}(x_{a})\,,

(8.56)

with the following properties:

(1)

If $x\in\mathcal{B}^{\ell}_{j}$ then $r_{h}(x)>r_{0}(n,{\rm v},D)\cdot{\rm diam}(\mathcal{B}^{\ell}_{j})$ .
(2)

Each neck $\mathcal{N}^{\ell}_{j}$ is diffeomorphic to $\mathds{R}\times S^{3}/\Gamma^{\ell}_{j}$ for some $\Gamma^{\ell}_{j}<O(4)$ .
(3)

$\mathcal{N}^{\ell}_{j}\cap\mathcal{B}^{\ell}_{j}$ are diffeomorphic to $\mathds{R}\times S^{3}/\Gamma^{\ell}_{j}$ . $\mathcal{N}^{\ell}_{j}\cap\mathcal{B}^{\ell-1}_{j^{\prime}}$ are either empty or diffeomorphic to $\mathds{R}\times S^{3}/\Gamma^{\ell}_{j}$ .
(4)

$N_{\ell}\leq N(n,{\rm v},D)$ .
(5)

If $\mathcal{N}^{\ell+1}_{a}\cap\mathcal{B}^{\ell}_{j}\neq\emptyset$ , then $|\Gamma^{\ell}_{a}|\leq|\Gamma^{\ell}_{j}|-1$ .
(6)

We have $r^{k}_{a}=r_{\alpha^{k}_{a}}$ with $T^{\delta}_{\alpha^{k}_{a}}=0$ , and $\mathcal{B}^{k}_{j}\cap B_{r^{k}_{a}}(x_{a})\subseteq A_{r^{k}_{a}/2,r^{k}_{a}}(x_{a})$ .
(7)

If $\beta^{k}_{a}$ is the largest integer such that $\mathcal{V}^{\delta}_{r_{\beta^{k}_{a}}/4}(x_{a})\geq\ln|\Gamma^{k}_{a}|-\delta$ , then for every $x\in B_{2r_{\beta^{k}_{a}}}(x_{a})$ we have $\mathcal{V}^{\delta}_{r_{\beta^{k}_{a}}/8}(x)<\ln|\Gamma^{k}_{a}|-\delta$ .

Before building the inductive decomposition, let us note that once we have it, we will have finished the proof. In fact, all we really need to see is that for some $k\leq k(n,{\rm v},D)$ , there are no balls $\{B_{r^{k}_{a}}(x_{a})\}$ in the decomposition. To see this, observe that by the lower volume bound we have the upper order bound $|\Gamma^{2}_{j}|\leq C({\rm v},D)$ . By condition (5) above we have by iteration that for each $j$ that there is some $j_{2}$ such that

\displaystyle 0\leq|\Gamma^{k}_{j}|\leq|\Gamma^{2}_{j_{2}}|-(k-2)\leq C({\rm v},D)-(k-2)\,,

(8.57)

and in particular this immediately implies the upper bound

\displaystyle k\leq k({\rm v},D)\,.

(8.58)

To prove the inductive decomposition, we begin by noting that (8.55) provides the basic case. So let us assume that the decomposition has been constructed for some $k$ , and let us build the decomposition for $k+1$ .

First, we use condition (7) and Lemma 8.8 to see that there exists an open set

\displaystyle A_{r_{\beta^{k}_{a}}/2,2r_{\alpha^{k}_{a}}}(x_{a})\subseteq\mathcal{N}^{k+1}_{a}\subseteq A_{(1-\epsilon)r_{\beta^{k}_{a}}/2,2(1+\epsilon)r_{\alpha^{k}_{a}}}(x_{a})\,,

(8.59)

and a diffeomorphism $\Phi^{k+1}_{a}:\mathcal{N}^{k+1}_{a}\to A_{r_{\beta^{k}_{a}}/2,2r_{\alpha^{k}_{a}}}(0)$ with $0\in\mathds{R}^{4}/\Gamma^{k}_{a}$ . By Lemma 8.9, there exists a radius $r_{a}=\bar{r}({\rm v},\delta)r_{\beta^{k}_{a}}$ such that

\displaystyle\mathcal{V}^{\delta}_{r_{a}}(x)<\ln\big(|\Gamma^{k+1}_{a}|-1\big)+\delta\,,

(8.60)

for every $x\in B_{2r_{\beta^{k}_{a}}}(x_{a})$ .

Pick some efficient covering $\{B_{r_{a}}(x_{aj}\}$ of $B_{2r_{\beta^{k}_{a}}}(x_{a})$ such that the balls in $\{B_{r_{a}/4}(x_{aj}\}$ disjoint. Now apply Lemma 8.10 to each ball $\{B_{r_{a}}(x_{aj}\}$ in order to construct a collection of balls $\{B_{r^{k+1}_{ab}}(x^{k+1}_{ab})\}$ with $r^{k+1}_{ab}=r_{\alpha^{k+1}_{ab}}>\bar{r}({\rm v},\delta)r_{\beta^{k}_{a}}$ . Observe that since there are at most $N({\rm v},D)$ balls in the collection $\{B_{r_{a}/4}(x_{aj}\}$ , and the application of Lemma 8.10 produces at most $N({\rm v},D)$ balls for each of these, we have at most $N({\rm v},D)$ such balls in total.

If we put

\displaystyle\{\mathcal{B}^{k+1}_{a}\ B_{2r_{\beta^{k}_{a}}}(x_{a})\setminus\cup B_{r_{\alpha^{k+1}_{ab}}}(x_{ab})\,,

(8.61)

we see that $\mathcal{B}^{k+1}_{a}$ and the collection $\{B_{r^{k+1}_{ab}}(x^{k+1}_{ab})\}$ satisfy the inductive conditions. Specifically, what is left to check is condition (5). However, by construction, we have

\displaystyle\ln(|\Gamma^{k}_{a}|-1)+\delta>\mathcal{V}^{\delta}_{r^{k+1}_{ab}}(x_{ab})\geq\ln|\Gamma^{k+1}_{ab}|-\delta\,,

(8.62)

which for $\delta({\rm v})$ sufficiently small implies $|\Gamma^{k+1}_{aj}|<|\Gamma^{k}_{a}|$ . In particular, the decomposition

\displaystyle M^{n}\equiv\mathcal{B}^{1}\cup\bigcup_{j_{2}=1}^{N_{2}}\mathcal{N}^{2}_{j_{2}}\bigcup_{j_{2}=1}^{N_{2}}\mathcal{B}^{2}_{j_{2}}\cup\cdots\cup\bigcup_{j_{k}=1}^{N_{k}}\mathcal{N}^{k}_{j_{k}}\bigcup_{j_{k}=1}^{N_{k}}\mathcal{B}^{k}_{j_{k}}\cup\bigcup_{a=1}^{N_{k+1}}\mathcal{N}^{k+1}_{a}\bigcup\mathcal{B}^{k+1}_{a}\bigcup B_{2r_{\alpha^{k+1}_{ab}}}(x_{ab})\,,

(8.63)

satisfies the inductive hypothesis as well, which completes the proof.

∎

Now that we have constructed the bubble tree in Theorem 8.11 let us finish the proof of Theorem 1.4:

[01ZT]

Proof of Theorem 1.4.

Let $M^{4}$ satisfy $|{\rm Ric}_{M^{4}}|\leq 3$ , ${\rm Vol}(M)>{\rm v}>0$ and ${\rm diam}(M^{4})\leq D$ . Then using Theorem 8.11, we can write

\displaystyle M^{4}\equiv\mathcal{B}^{1}\cup\bigcup_{j_{2}=1}^{N_{2}}\mathcal{N}^{2}_{j_{2}}\cup\bigcup_{j_{2}=1}^{N_{2}}\mathcal{B}^{2}_{j_{2}}\cup\cdots\cup\bigcup_{j_{k}=1}^{N_{k}}\mathcal{N}^{k}_{j_{k}}\cup\bigcup_{j_{k}=1}^{N_{k}}\mathcal{B}^{k}_{j_{k}}\,.

(8.64)

First we will analyze each body region $\mathcal{B}^{k}_{j}$ . Indeed, by (1) and theorem 8.2, it follows that there are at most $C({\rm v},D)$ -diffeomorphism types for each $\mathcal{B}^{k}_{j}$ . By (4), there are at most $C({\rm v},D)$ such body regions, and by (2) and (3), there are at most $C({\rm v},D)$ diffeomorphism types that can arise by gluing them together, which proves the theorem. ∎

[01ZU]

8.5. $L^{2}$ Curvature Estimates

We begin with the following, whose proof is essentially the same as that of Theorem 1.4 of the previous subsection:

[01ZV]

Theorem 8.12.

There exists $\delta({\rm v})>0$ such that if $M^{4}$ satisfies $|{\rm Ric}_{M^{4}}|<2\delta$ , ${\rm Vol}(B_{1}(p))>{\rm v}>0$ , and $T^{\delta}_{0}(p)=0$ , then there exists $B_{1}(p)\subseteq U\subseteq B_{2}(p)$ such that $U$ has at most $C({\rm v})$ diffeomorphism types. Further, $U$ can be chosen so that it’s boundary $\partial U$ is diffeomorphic to $S^{3}/\Gamma$ and satisfies the second fundamental form estimate $|A|\leq C(v)$ .

[01ZW]

Proof.

The proof is the same as that of Theorem 1.4, except for the second fundamental form estimate on the boundary. To see this estimate, we use $T^{\delta}_{0}(p)=0$ and Theorem 8.3 to find a diffeomorphism $\Phi:A_{1/2,2}(0)\to B_{1}(p)$ onto its image, such that if $g_{ij}=\Phi^{*}g$ is the pullback metric then

\displaystyle||g_{ij}-\delta_{ij}||_{C^{0}}+||\partial_{k}g_{ij}||_{C^{0}}<\epsilon\,.

(8.65)

In particular, we can choose $U$ so that its boundary is $\partial U=\partial B_{3/2}(0)$ in these coordinates. The $C^{1}$ estimates on $g$ give rise to the appropriate second fundamental form estimates on $\partial U$ . ∎

With this in hand we are in a position to finish the proof of Theorem 1.5:

[01ZX]

Proof of Theorem 1.5.

Let $M^{4}$ satisfy $|{\rm Ric}_{M^{4}}|\leq 3$ and ${\rm Vol}(B_{1}(p))>{\rm v}>0$ . Using volume monotonicity, we have for every $x\in B_{1}(p)$ and $r\leq 1$ ,

\displaystyle{\rm Vol}(B_{r}(x))\geq\frac{{\rm Vol}_{-1}(B_{r})}{{\rm Vol}_{-1}(B_{2})}{\rm Vol}(B_{2}(x))\geq c(n){\rm Vol}(B_{1}(p))\,r^{4}\geq c{\rm v}\,r^{4}\,.

(8.66)

Let $\delta({\rm v})$ be as in Theorem 8.12. By Lemma 8.5, we have that for each $x\in B_{1}(p)$ , there exists a radius, $r_{\alpha_{x}}=2^{-\alpha_{x}}\in[C({\rm v})\delta^{3},\delta^{2}]$ , such that $T^{\delta}_{\alpha_{x}}(x)=0$ . Let $\{B_{r_{i}}(x_{i})\}$ be a subcovering such that the balls in $\{B_{r_{i}/4}(x_{i})\}$ are disjoint, where $r_{i}=r_{\alpha_{x_{i}}}$ . Since $r_{i}>\bar{r}({\rm v})$ , we have by the usual doubling estimates that there are at most $C({\rm v})$ balls in this covering.

Note that, for each ball $B_{r_{i}}(x_{i})$ , we can apply Theorem 8.12 in order to get a subset $U_{i}\supseteq B_{r_{i}}(x_{i})$ with bounded diffeomorphism type and uniform boundary control. Now recall in dmiension $4$ , the Chern-Guass-Bonnet formula can be written as

\displaystyle\chi(U_{i})=\frac{1}{32\pi^{2}}\int_{U_{i}}|{\rm Rm}|^{2}-4|{\rm Ric}|^{2}+R^{2}+\int_{\partial U_{i}}\Psi\,,

(8.67)

where $\Psi=\Psi(A)$ is a function of the second fundamental form. By reorganizing, we obtain the bound

	$\displaystyle\int_{U_{i}}\|{\rm Rm}\|^{2}$	$\displaystyle\leq 32\pi^{2}\|\chi(U_{i})\|+4\int_{U_{i}}\|{\rm Ric}\|^{2}+C\int_{U_{i}}\|\Psi\|\,,$
		$\displaystyle\leq C({\rm v})\,,$		(8.68)

where we have used the bound on the diffeomorphism type, the Ricci bound, and the second fundamental form bound from Theorem 8.12. By summing over $i$ , we get

\displaystyle\fint_{B_{1}(p)}|{\rm Rm}|^{2}\leq C({\rm v})\sum\int_{U_{i}}|{\rm Rm}|^{2}\leq C({\rm v})\,,

(8.69)

as claimed. ∎

[01ZY]

9. Conjectures

In this section, we briefly remark on some possible extensions of the results of this paper. To begin with, we recall that one of the main applications of this paper was to combine the codimension $4$ estimates of Theorem 1.1 with the ideas of quantitative stratification in order to show for all $q<2$ that $\fint_{B_{1}(p)}|{\rm Rm}|^{q}$ is uniformly bounded when $M^{n}$ is a noncollapsed manifold with bounded Ricci curvature. Furthermore, in dimension $4$ we were able to improve this to show a bound on $\fint_{B_{1}(p)}|{\rm Rm}|^{2}$ . We conjecture that this holds in any dimension.

[01ZZ]

Conjecture 9.1.

There exists $C=C(n,{\rm v})>0$ such that if $M^{n}$ satisfies $|{\rm Ric}_{M^{n}}|\leq n-1$ and ${\rm Vol}(B_{1}(p))>{\rm v}>0$ , then

\displaystyle\fint_{B_{1}(p)}|{\rm Rm}|^{2}\leq C\,.

(9.1)

In a different direction, another main result of the paper was to show that in dimension $4$ , noncollapsed manifolds with bounded diameter and Ricci curvature have finite diffeomorphism type. In higher dimensions, this is too much to hope for; see for instance [HN14] where noncollapsed Calabi-Yau manifolds of real dimension $\geq 6$ are constructed with unbounded third Betti number. Nonetheless, we conjecture that under the assumption of bounded Ricci curvature, one should expect a bound on the second Betti number.

[0200]

Conjecture 9.2.

There exists $C=C(n,{\rm v},D)$ such that if $M^{n}$ satisfy $|{\rm Ric}_{M^{n}}|\leq n-1$ , ${\rm diam}(M^{n})\leq D$ , and ${\rm Vol}(B_{1}(p))>v>0$ , then $b_{2}(M^{n})\leq C$ .

Note that examples of Menguy and Perelman show that it is actually necessary to assume a $2$ -sided bound on the Ricci tensor; see [Men2000].

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$\displaystyle\|u(B_{\eta})\|$	$\displaystyle\leq\sum\|u(B_{6s_{j}}(x_{j}))\|\leq\sum({6s_{j}})^{-2}\mu(B_{6s_{j}}(x_{j}))$
	$\displaystyle\leq C(n)\sum s_{j}^{-2}\mu(B_{s_{j}}(x_{j}))\leq C\eta^{-1}\sum\int_{B_{s_{j}}}\|\Delta\|\omega^{\ell}\|\|$
	$\displaystyle\leq C\eta^{-1}\int_{B_{2}}\|\Delta\|\omega^{\ell}\|\,\|\leq\delta^{\prime}\,,$	(1.22)

$\displaystyle C(n)\fint_{B_{2r}(x)}\|\,\|\nabla u\|^{2}-c\|$	$\displaystyle\geq\fint_{B_{2r}(x)}\|\Delta\varphi\|\cdot\|\nabla u\|^{2}-c\|$	(3.8)
	$\displaystyle\geq\left\|\fint_{B_{2r}(x)}\varphi\Delta(\|\nabla u\|^{2}-c)\right\|$
	$\displaystyle=\left\|\fint_{B_{2}(x)}\varphi(\|\nabla^{2}u\|^{2}+{\rm Ric}(\nabla u,\nabla u))\right\|$
	$\displaystyle\geq\fint_{B_{r}(x)}\|\nabla^{2}u\|^{2}-\delta\|\nabla u\|^{2}\,,$

$\displaystyle\int\varphi\frac{\|\nabla^{2}u\|^{2}}{\|\nabla u\|^{1+\alpha}}$	$\displaystyle\leq C(n,\alpha)\int\Big(\varphi\Delta\|\nabla u\|^{1-\alpha}+\varphi\delta\|\nabla u\|^{1-\alpha}\Big)\,,$
	$\displaystyle\leq C(n,\alpha)\int\Delta\varphi\Big(\|\nabla u\|^{1-\alpha}-\fint_{B_{2}(p)}\|\nabla u\|^{1-\alpha}\Big)+C(n,\alpha)\delta\int_{B_{2}(p)}\|\nabla u\|^{1-\alpha}\,,$
	$\displaystyle\leq C(n,\alpha)\int_{B_{2}(p)}\Big\|\|\nabla u\|^{1-\alpha}-\fint_{B_{2}(p)}\|\nabla u\|^{1-\alpha}\Big\|+C(n,\alpha)\delta\int_{B_{2}(p)}\|\nabla u\|^{1-\alpha}\,,$	(3.17)

	$\displaystyle\fint_{B_{3/2}(p)}\frac{\|\nabla^{2}u\|^{2}}{\|\nabla u\|^{1+\alpha}}$	$\displaystyle\leq C(n)\fint_{B_{2}(p)}\varphi\frac{\|\nabla^{2}u\|^{2}}{\|\nabla u\|^{1+\alpha}}$
		$\displaystyle\leq C(n,\alpha)\fint_{B_{2}(p)}\Big\|\|\nabla u\|^{1-\alpha}-\fint_{B_{2}(p)}\|\nabla u\|^{1-\alpha}\Big\|+C(n,\alpha)\delta\fint_{B_{2}(p)}\|\nabla u\|^{1-\alpha}\leq\epsilon\,,$		(3.18)

$\displaystyle\fint_{B_{2}(p)}\varphi\Big\|$	$\displaystyle\Delta\|\omega^{\ell}\|-\langle\sum_{a,b}du_{a_{1}}\wedge\nabla^{j}(du_{a_{a}})\wedge\cdots\wedge\nabla_{j}(du_{a_{b}})\wedge\cdots\wedge du_{a_{k}},\frac{\omega^{\ell}}{\|\omega^{\ell}\|}\rangle+C(n)\delta\|\omega^{\ell}\|\Big\|$
	$\displaystyle=\fint_{B_{2}(p)}\varphi\Big(\Delta\|\omega^{\ell}\|-\langle\sum_{a,b}du_{a_{1}}\wedge\nabla^{j}(du_{a_{a}})\wedge\cdots\wedge\nabla_{j}(du_{a_{b}})\wedge\cdots\wedge du_{a_{k}},\frac{\omega^{\ell}}{\|\omega^{\ell}\|}\rangle+C(n)\delta\|\omega^{\ell}\|\Big)$
	$\displaystyle\leq\fint_{B_{2}(p)}\Delta\varphi\Big(\|\omega^{\ell}\|-1\Big)+\sum\fint_{B_{2}(p)}\|\nabla^{2}u_{a_{j}}\|^{2}+C\delta\fint_{B_{2}(p)}\|\omega^{\ell}\|\,,$
	$\displaystyle\leq C(n)\fint\big\|\|\omega^{\ell}\|-1\big\|+C\delta<\frac{\epsilon}{3}\,,$	(3.21)

Regularity of Einstein Manifolds and the Codimension 44 Conjecture

Abstract.

1. Introduction

Theorem 1.1.

Definition 1.2.

Theorem 1.3.

Remark 1.1.

Theorem 1.4.

Theorem 1.5.

Remark 1.2.

1.1. Outline of the proof Theorem 1.1, the codimension 4 conjecture

Definition 1.6.

Lemma 1.7 ([ChCo1]).

Theorem 1.8.

Theorem 1.9.

Definition 1.10.

Theorem 1.11.

2. Background and Preliminaries

2.1. Stratification of Limit Spaces

Definition 2.1.

2.2. ϵ\epsilon-Regularity Theorems

Definition 2.2.

Theorem 2.3 ([A90], [ChCo1]).

2.3. Examples

Example 2.1.

Example 2.2.

Example 2.3.

3. Proof of the Transformation Theorem

3.1. Higher Order Estimates

Proof of Theorem 1.9.

3.2. Proof of the Transformation theorem

Proof of Theorem 1.11:

Lemma 3.1.

Remark 3.1.

Remark 3.2.

4. Proof of Theorem 1.8, the Slicing Theorem

Lemma 4.1.

Proof.

Lemma 4.2.

Proof.

5. Codimension 44 Regularity of Singular Limits

5.1. Nonexistence of Codimension 22 Singularities

Theorem 5.1 ((n−2)(n-2)-Symmetric Limits).

Proof of Theorem 5.1.

Corollary 5.2.

Proof.

5.2. Nonexistence of Codimension 33 singularities

Theorem 5.3 ((n−3)(n-3)-Symmetric Limits).

Proof.

5.3. Proof of Hausdorff Estimates of Theorem 1.1

6. The ϵ\epsilon-regularity Theorem

Theorem 6.1.

Proof.

7. Quantitative Stratification and Effective Estimates

Definition 7.1.

Definition 7.2.

Theorem 7.3 (Quantitative Stratification,[ChNa13]).

7.1. Proof of Theorem 1.3

Proof.

7.2. LqL^{q}-Estimates for Harmonic Functions on Einstein Manifolds

Theorem 7.4.

Proof.

8. Improved Estimates in Dimension 4

8.1. Diffeomorphisms and Harmonic Radius

Theorem 8.1.

Theorem 8.2.

8.2. Annulus Estimates

Theorem 8.3.

Proof.

8.3. Regularity Scale Estimates

Definition 8.4.

Remark 8.1.

Lemma 8.5.

Proof.

Corollary 8.6.

Proof.

Proof of Estimate (1.9) of Theorem 1.5.

8.4. Finite Diffeomorphism Type

Definition 8.7.

Lemma 8.8.

Regularity of Einstein Manifolds
and the Codimension $4$ Conjecture

2.2. $\epsilon$ -Regularity Theorems

5. Codimension $4$ Regularity of Singular Limits

5.1. Nonexistence of Codimension $2$ Singularities

Theorem 5.1 ( $(n-2)$ -Symmetric Limits).

5.2. Nonexistence of Codimension $3$ singularities

Theorem 5.3 ( $(n-3)$ -Symmetric Limits).

6. The $\epsilon$ -regularity Theorem

7.2. $L^{q}$ -Estimates for Harmonic Functions on Einstein Manifolds

8.5. $L^{2}$ Curvature Estimates