8.2. Annulus Estimates [01Z7]

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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.

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)

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. ∎