Rectifiability of Singular Sets in Noncollapsed Spaces with Ricci Curvature bounded below (1805.07988v1)
Abstract: This paper is concerned with the structure of Gromov-Hausdorff limit spaces $(Mn_i,g_i,p_i)\stackrel{d_{GH}}{\longrightarrow} (Xn,d,p)$ of Riemannian manifolds satisfying a uniform lower Ricci curvature bound $Rc_{Mn_i}\geq -(n-1)$ as well as the noncollapsing assumption $Vol(B_1(p_i))>v>0$. In such cases, there is a filtration of the singular set, $S_0\subset S_1\cdots S_{n-1}:= S$, where $Sk:= {x\in X:\text{ no tangent cone at $x$ is }(k+1)\text{-symmetric}}$; equivalently no tangent cone splits off a Euclidean factor $\mathbb{R}{k+1}$ isometrically. Moreover, by \cite{ChCoI}, $\dim Sk\leq k$. However, little else has been understood about the structure of the singular set $S$. Our first result for such limit spaces $Xn$ states that $Sk$ is $k$-rectifiable. In fact, we will show that for $k$-a.e. $x\in Sk$, {\it every} tangent cone $X_x$ at $x$ is $k$-symmetric i.e. that $X_x= \mathbb{R}k\times C(Y)$ where $C(Y)$ might depend on the particular $X_x$. We use this to show that there exists $\epsilon=\epsilon(n,v)$, and a $(n-2)$-rectifible set $S{n-2}_\epsilon$, with finite $(n-2)$-dimensional Hausdorff measure $H{n-2}(S_\epsilon{n-2})<C(n,v)$, such that $Xn\setminus S{n-2}_\epsilon$ is bi-H\"older equivalent to a smooth riemannian manifold. This improves the regularity results of \cite{ChCoI}. Additionally, we will see that tangent cones are unique of a subset of Hausdorff $(n-2)$ dimensional measure zero. Our analysis is based on several new ideas, including a sharp cone-splitting theorem and a geometric transformation theorem, which will allow us to control the degeneration of harmonic functions on these neck regions.