(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. (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.