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Let $\Sigma$ be a complete Riemannian manifold of nonnegative Riccicurvature. We prove a Liouville-type theorem: every smooth solution $u$ tominimal hypersurface equation on $\Sigma$ is a constant provided $u$ hassublinear growth for its negative part. Here, the sublinear growth condition issharp. Our proof relies on a gradient estimate for minimal graphs over $\Sigma$with small linear growth of the negative parts of graphic functions viaiteration.

Liouville theorem for minimal graphs over manifolds of nonnegative Ricci curvature