Area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below

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 $\ge -n{\kappa }^2$ on $B_{1+{\kappa }^{\prime }}(p_i)$ for constants $\kappa \ge 0$, ${\kappa }^{\prime }>0$, 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.

39 pages

References in corpus (1)

• Poincaré inequality on minimal graphs over manifolds and applications