Integral bounds on curvature, elliptic estimates and rectifiability of singular sets
Let \((Y,y)\) be a pointed Gromov-Hausdorff limit of a pointed sequence of Riemannian manifolds \((M_i,m_i)\) satisfying: (1) \(\text{Ric}_{M_i^n} \geq -(n-1)\), (2) \(\text{Vol}(B_1(m_i)) \geq c_{0 > 0}\), (3) \(\int_{B_r}(m_i) |R|^p \leq C(r)\) [\textit{J. Cheeger} and \textit{T. H. Colding}, J. Differ. Geom. 46, No. 3, 406--480 (1997; Zbl 0902.53034) and J. Differ. Geom. 54, No. 1, 13--35 (2000; Zbl 1027.53042)]. A metric space \(W\) is called \(d\)-rectifiable if \(0 < \mathcal {H}^d(W) < \infty \) and there exists a countable collection of subsets \(C_j\) with \(\mathcal{H}^d(W - \bigcup _j C_j) = 0\) such that each \(C_j\) is bi-Lipschitz to a subset of \({\mathbb R}^d\), where \(\mathcal {H}^d\) denotes the d-dimensional Hausdorff measure. This paper is an additional step in the program initiated by \textit{J. Cheeger} and \textit{T. H. Colding} [Geom. Funct. Anal. 12, No. 5, 873--914 (2002; Zbl 1030.53046)] and contains the following sections: elliptic estimates, the divergence theorem and sets of small capacity, decoupled estimates, \(\varepsilon\)-regularity theorems (real case), \(\varepsilon\) regularity and gap theorems (Kähler case), applications of \(\varepsilon\)-regularity, existence of tangent cones of orbifold type, \(\varepsilon\)-regularity and gap theorems (the nonexceptional case), inverting integer \(p\). The main result here establishes the \((n - 2p)\)-rectiafibility of compact subsets of the singular set for spaces satisfying the conditions (1)--(3) above, with \(M_i\) Kähler of real dimension \(n\).
scientific article; zbMATH DE number 1971825