Poincar\'e inequality on minimal graphs over manifolds and applications

Let $B_2(p)$ be an $n$-dimensional smooth geodesic ball with Ricci curvature $\geq-(n-1)\kappa^2$ for some $\kappa\geq0$. We establish the Sobolev inequality and the uniform Neumann-Poincar\'e inequality on each minimal graph over $B_1(p)$ by combining Cheeger-Colding theory and the current theory from geometric measure theory, where the constants in the inequalities only depends on $n$, $\kappa$, the lower bound of the volume of $B_1(p)$. As applications, we derive gradient estimates and a Liouville theorem for a minimal graph $M$ over a smooth complete noncompact manifold $\Sigma$ of nonnegative Ricci curvature and Euclidean volume growth. Furthermore, we can show that any tangent cone of $\Sigma$ at infinity splits off a line isometrically provided the graphic function of $M$ admits linear growth.