Liouville theorem for minimal graphs over manifolds of nonnegative Ricci curvature

Let $\mathrm{\Sigma }$ be a complete Riemannian manifold of nonnegative Ricci curvature. We prove a Liouville-type theorem: every smooth solution $u$ to minimal hypersurface equation on $\mathrm{\Sigma }$ is a constant provided $u$ has sublinear growth for its negative part. Here, the sublinear growth condition is sharp. Our proof relies on a gradient estimate for minimal graphs over $\mathrm{\Sigma }$ with small linear growth of the negative parts of graphic functions via iteration.