Minimal hypersurfaces in manifolds of Ricci curvature bounded below

Abstract

Abstract In this paper, we study the angle estimate of distance functions from minimal hypersurfaces in manifolds of Ricci curvature bounded from below using Colding’s method in [T. H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 1997, 3, 477–501]. With Cheeger–Colding theory, we obtain the Laplacian comparison for limits of distance functions from minimal hypersurfaces in the version of Ricci limit space. As an application, if a sequence of minimal hypersurfaces converges to a metric cone C ⁢ Y × ℝ n - k {CY\times\mathbb{R}^{n-k}} ( 2 ≤ k ≤ n {2\leq k\leq n} ) in a non-collapsing metric cone C ⁢ X × ℝ n - k {CX\times\mathbb{R}^{n-k}} obtained from ambient manifolds of almost nonnegative Ricci curvature, then we can prove a Frankel property for the cross section Y of CY. Namely, Y has only one connected component in X.