Area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below

Abstract

Abstract In this paper, we study area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below with Cheeger–Colding theory. Let N i {N_{i}} be a sequence of smooth manifolds with Ricci curvature ≥ - n ⁢ κ 2 {\geq-n\kappa^{2}} on B 1 + κ ′ ⁢ ( p i ) {B_{1+\kappa^{\prime}}(p_{i})} for constants κ ≥ 0 {\kappa\geq 0} , κ ′ > 0 {\kappa^{\prime}>0} , and volume of B 1 ⁢ ( p i ) {B_{1}(p_{i})} has a positive uniformly lower bound. Assume B 1 ⁢ ( p i ) {B_{1}(p_{i})} converges to a metric ball B 1 ⁢ ( p ∞ ) {B_{1}(p_{\infty})} in the Gromov–Hausdorff sense. For a sequence of area-minimizing hypersurfaces M i {M_{i}} in B 1 ⁢ ( p i ) {B_{1}(p_{i})} with ∂ ⁡ M i ⊂ ∂ ⁡ B 1 ⁢ ( p i ) {\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 ∞ {M_{\infty}} of M i {M_{i}} is area-minimizing in B 1 ⁢ ( p ∞ ) {B_{1}(p_{\infty})} provided B 1 ⁢ ( p ∞ ) {B_{1}(p_{\infty})} is a smooth Riemannian manifold. By blowing up argument, we get sharp dimensional estimates for the singular set of M ∞ {M_{\infty}} in ℛ {\mathcal{R}} , and 𝒮 ∩ M ∞ {\mathcal{S}\cap M_{\infty}} . Here, ℛ {\mathcal{R}} and 𝒮 {\mathcal{S}} are the regular and singular parts of B 1 ⁢ ( p ∞ ) {B_{1}(p_{\infty})} , respectively.