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. (1)

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

2. (2)

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

3. (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

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

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

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

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

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.

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

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

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

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.

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

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