Poincaré inequality on minimal graphs over manifolds and applications

Abstract

Let B2(p) be an n-dimensional smooth geodesic ball with Ricci curvature ≥ −(n − 1)κ 2 for some κ ≥ 0. We establish the Sobolev inequality and the uniform Neumann-Poincaré inequality on each minimal graph over B1(p) by combining Cheeger-Colding theory and the current theory from geometric measure theory, where the constants in the inequalities only depends on n, κ, the lower bound of the volume of B1(p). As applications, we derive gradient estimates and a Liouville theorem for a minimal graph M over a smooth complete noncompact manifold Σ of nonnegative Ricci curvature and Euclidean volume growth. Furthermore, we can show that any tangent cone of Σ at infinity splits off a line isometrically provided the graphic function of M admits linear growth.