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Rectifiability of singular sets of noncollapsed limit spaces with Ricci curvature bounded below. (English) Zbl 07331714
Summary: This paper is concerned with the structure of Gromov-Hausdorff limit spaces $$(M^n_i,g_i,p_i)\stackrel{d_{GH}}{\longrightarrow} (X^n,d,p)$$ of Riemannian manifolds satisfying a uniform lower Ricci curvature bound $$\text{Ric}_{M^n_i}\geq -(n-1)$$ as well as the noncollapsing assumption $$\text{Vol}(B_1(p_i))>\text{v}>0$$. In such cases, there is a filtration of the singular set, $$S^0\subset S^1\cdots S^{n-1}:= S$$, where $$S^k:= \{x\in X:\text{ no tangent cone at }x \text{ is }(k+1)\text{-symmetric}\}$$. Equivalently, $$S^k$$ is the set of points such that no tangent cone splits off a Euclidean factor $$\mathbb{R}^{k+1}$$. It is classical from Cheeger-Colding that the Hausdorff dimension of $$S^k$$ satisfies $$\dim S^k\leq k$$ and $$S=S^{n-2}$$, i.e., $$S^{n-1}\setminus S^{n-2}=\emptyset$$. However, little else has been understood about the structure of the singular set $$S$$.
Our first result for such limit spaces $$X^n$$ states that $$S^k$$ is $$k$$-rectifiable for all $$k$$. In fact, we will show for $$\mathcal H^k$$-a.e. $$x\in S^k$$ that 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 $$X_x$$. Here $$\mathcal{H}^k$$ denotes the $$k$$-dimensional Hausdorff measure. As an application we show for all $$0<\epsilon<\epsilon(n,\text{v})$$ there exists an $$(n-2)$$-rectifiable closed set $$S^{n-2}_\epsilon$$ with $$\mathcal{H}^{n-2}(S_{\epsilon}^{n-2})<C(n,\text{v},\epsilon)$$, such that $$X^n\setminus S^{n-2}_\epsilon$$ is $$\epsilon$$-bi-Hölder equivalent to a smooth Riemannian manifold. Moreover, $$S=\bigcup_\epsilon S^{n-2}_\epsilon$$. As another application, we show that tangent cones are unique $$\mathcal H^{n-2}$$-a.e.
In the case of limit spaces $$X^n$$ satisfying a $$2$$-sided Ricci curvature bound $$|\text{Ric}_{M^n_i}|\leq n-1$$, we can use these structural results to give a new proof of a conjecture from Cheeger-Colding stating that $$S$$ is $$(n-4)$$-rectifiable with uniformly bounded measure. We can also conclude from this structure that tangent cones are unique $$\mathcal H^{n-4}$$-a.e.
Our analysis builds on the notion of quantitative stratification introduced by Cheeger-Naber, and the neck region analysis developed by Jiang-Naber-Valtorta. Several new ideas and new estimates are required, 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.
##### MSC:
 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 58A35 Stratified sets 35A21 Singularity in context of PDEs
##### Keywords:
Ricci; curvature; stratification; rectifiability
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