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Rectifiability of singular sets of noncollapsed limit spaces with Ricci curvature bounded below. (English) Zbl 1469.53083

Authors’ abstract: 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
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References:

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