Liouville theorem for minimal graphs over manifolds of nonnegative Ricci curvature

TL;DR

The paper proves a strong Liouville-type theorem for minimal graphs over complete non-compact manifolds $\Sigma$ with nonnegative Ricci curvature: any smooth solution $u$ to the minimal graph equation with sublinear growth of its negative part must be constant. The key technical advance is a gradient estimate for minimal graphs under small linear negative-growth, achieved through an iteration on the volume function $v=\sqrt{1+|Du|^2}$ that relies on integral bounds for $\log v$ and Sobolev-type inequalities on $\Sigma$, followed by a De Giorgi–Nash–Moser type argument to obtain mean-value and gradient bounds. The results extend Liouville-type rigidity from Euclidean spaces to geometric settings with Ricci curvature nonnegativity and highlight the sharpness of the sublinear condition. An appendix barbs the barrier/Harnack framework that ensures the global rigidity by ruling out non-constant solutions under the stated growth constraints. Overall, the work connects asymptotic behavior of minimal graphs to geometric-analytic inequalities and contributes a sharp gradient-controlled Liouville theorem in nonnegatively curved manifolds.

Abstract

Let $Σ$ be a complete Riemannian manifold of nonnegative Ricci curvature. We prove a Liouville-type theorem: every smooth solution $u$ to minimal hypersurface equation on $Σ$ 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 $Σ$ with small linear growth of the negative parts of graphic functions via iteration.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4