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).