Let $\Sigma$ be a complete Riemannian manifold of nonnegative Ricci curvature. We prove a Liouville-type theorem: every smooth solution $u$ to minimal hypersurface equation on $\Sigma$ 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 $\Sigma$ with small linear growth of the negative parts of graphic functions via iteration.
Capacity for minimal graphs over manifolds and the half-space property
Explicit sharp constants in Sobolev inequalities on Riemannian manifolds with nonnegative Ricci curvature
Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature
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