Abstract: Let N be an (n+ 1)-dimensional complete manifold of Ricci curvature ≥ ?n, and B2(p) be the
geodesic ball in N. Let M be an area-minimizing hypersurface in B2(p) with p ∈ M and ?M ? ?B2(p). In this
talk, we will discuss the Sobolev and Neumann Poincar?e inequalities on M ∩ B1(p). As an application, we
get the gradient estimates for the solutions of the minimal hypersurface equation on an n-manifold with Ric ≥ ?(n?1).
