On minimal graphs of sublinear growth over manifolds with non-negative Ricci curvature

TL;DR

This work addresses Liouville-type rigidity for entire solutions of the minimal hypersurface equation on complete manifolds with non-negative Ricci curvature. Using a novel gradient-control method based on integral estimates and an auxiliary function $z=W e^{-Cu}$, the authors prove that sublinear growth of the negative part $u_-$ (specifically $u_-(x)=\mathcal{O}(r(x)/\log r(x))$) forces $u$ to be constant, extending Bernstein-type results beyond positive graphs. The approach avoids additional geometric assumptions and yields near-sharp growth thresholds, with further results showing decay of the gradient measure on superlevel sets under $u=o(r)$. Collectively, the paper advances understanding of global behavior and rigidity of minimal graphs over manifolds with $\mathrm{Ric}\ge0$, and connects to splitting-type phenomena and tangent-cone analyses at infinity.

Abstract

We prove that entire solutions of the minimal hypersurface equation \[ \mathrm{div}\left(\frac{Du}{\sqrt{1+|Du|^2}}\right) = 0 \] on a complete manifold with $\mathrm{Ric} \ge 0$, whose negative part grows like $\mathcal{O}(r/\log r)$ ($r$ the distance from a fixed origin), are constant. This extends the Bernstein Theorem for entire positive minimal graphs established in recent years. The proof depends on a new technique to get gradient bounds by means of integral estimates, which does not require any further geometric assumption on $M$.

Theorems & Definitions (25)

• Theorem A
• Remark 1
• Theorem 2: cgmr
• Theorem : cmmr,ding21
• Conjecture 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6