Area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below

Abstract

AbstractIn this paper, we study area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below with Cheeger–Colding theory. LetNi{N_{i}}be a sequence of smooth manifolds with Ricci curvature≥-n⁢κ2{\geq-n\kappa^{2}}onB1+κ′⁢(pi){B_{1+\kappa^{\prime}}(p_{i})}for constantsκ≥0{\kappa\geq 0},κ′>0{\kappa^{\prime}>0}, and volume ofB1⁢(pi){B_{1}(p_{i})}has a positive uniformly lower bound. AssumeB1⁢(pi){B_{1}(p_{i})}converges to a metric ballB1⁢(p∞){B_{1}(p_{\infty})}in the Gromov–Hausdorff sense. For a sequence of area-minimizing hypersurfacesMi{M_{i}}inB1⁢(pi){B_{1}(p_{i})}with∂⁡Mi⊂∂⁡B1⁢(pi){\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 limitM∞{M_{\infty}}ofMi{M_{i}}is area-minimizing inB1⁢(p∞){B_{1}(p_{\infty})}providedB1⁢(p∞){B_{1}(p_{\infty})}is a smooth Riemannian manifold. By blowing up argument, we get sharp dimensional estimates for the singular set ofM∞{M_{\infty}}inℛ{\mathcal{R}}, and𝒮∩M∞{\mathcal{S}\cap M_{\infty}}. Here,ℛ{\mathcal{R}}and𝒮{\mathcal{S}}are the regular and singular parts ofB1⁢(p∞){B_{1}(p_{\infty})}, respectively.