Area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below

TL;DR

This paper investigates the behavior of area-minimizing hypersurfaces in manifolds with Ricci curvature bounded below, establishing continuity of volume functions and sharp estimates for singular sets under Gromov-Hausdorff convergence.

Contribution

It proves the continuity of volume functions for area-minimizing hypersurfaces in Ricci-bounded manifolds and derives sharp estimates for their singular sets in limit spaces.

Findings

Volume functions are continuous under Gromov-Hausdorff convergence.

Limit hypersurfaces remain area-minimizing in smooth limit spaces.

Sharp dimensional estimates for singular sets are obtained.

Abstract

In this paper, we study area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below with Cheeger-Colding theory. Let $N_i$ be a sequence of smooth manifolds with Ricci curvature $\geq-n\kappa^2$ on $B_{1+\kappa'}(p_i)$ for constants $\kappa\ge0$, $\kappa'>0$, and volume of $B_1(p_i)$ has a positive uniformly lower bound. Assume $B_1(p_i)$ converges to a metric ball $B_1(p_\infty)$ in the Gromov-Hausdorff sense. For an area-minimizing hypersurface $M_i$ in $B_1(p_i)$ with $\partial M_i\subset\partial B_1(p_i)$, we prove the continuity for the volume function of area-minimizing hypersurfaces equipped with the induced Hausdorff topology. In particular, each limit $M_\infty$ of $M_i$ is area-minimizing in $B_1(p_\infty)$ provided $B_1(p_\infty)$ is a smooth Riemannian manifold. By blowing up argument, we get sharp dimensional estimates for the singular set of $M_\infty$ in $\mathcal{R}$, and $\mathcal{S}\cap M_\infty$. Here, $\mathcal{R}$, $\mathcal{S}$ are the regular and singular parts of $B_1(p_\infty)$, respectively.