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.

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

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

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

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

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

In particular,

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

Note that

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

In particular we have

Combining this with the integral estimate (3.28) we get

Now using the Bochner formula

we can estimate

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.

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

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

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

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

However in addition we also have

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

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

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

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

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

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. $\square$

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

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

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

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

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

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

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

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

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

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]

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

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

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

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

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

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

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 $x\in U_{j}\subseteq B_{R}(x_{j})$,

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 $x\in B_{2R}(x_{j})$