Minimal hypersurfaces in manifolds of Ricci curvature bounded below

TL;DR

This paper investigates the geometric properties of minimal hypersurfaces in manifolds with Ricci curvature bounded below, establishing angle estimates, Laplacian comparisons, and a Frankel property in Ricci limit spaces.

Contribution

It introduces new angle estimates and Laplacian comparison results for minimal hypersurfaces in Ricci limit spaces, and proves a Frankel property for the cross section of certain metric cones.

Findings

Established angle estimates for distance functions from minimal hypersurfaces.

Derived Laplacian comparison results in Ricci limit spaces.

Proved a Frankel property for the cross section of metric cones in Ricci limit spaces.

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 [13]. 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 $CY\times\mathbb{R}^{n-k}(2\leq k\leq n)$ in a non-collapsing metric cone $CX\times\mathbb{R}^{n-k}$ 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 $X$.