Minimal Surface Equation and Bernstein Property on RCD spaces

TL;DR

The paper extends classical Bernstein results to the non-smooth setting of $ extsf{RCD}(K,N)$ spaces by showing that a local graph-theoretic minimal surface equation implies harmonicity of the height function on the graph. The authors construct a metric-measure graph $\mathsf{G}(u)$ with a compatible Sobolev structure, prove that the height $u_g$ is harmonic on $\mathsf{G}(u)$, and derive a Harnack inequality via a Moser-type iteration adapted to this framework. This yields a Bernstein property in the non-smooth setting and provides oscillation estimates for minimal graphs, with applications to smooth weighted manifolds and products $M\times\mathbb{R}$ under non-negative Ricci curvature. The work combines blow-up analysis, tangent-ball arguments, and a novel product-gradient calculus on graphs to transfer elliptic estimates to the metric-measure context, broadening the scope of minimal surface theory beyond smooth manifolds.

Abstract

We show that if $(X,d,m)$ is an RCD(K,N) space and $u \in W^{1,1}_{loc}(X)$ is a solution of the minimal surface equation, then $u$ is harmonic on its graph (which has a natural metric measure space structure). If K=0 this allows to obtain an Harnack inequality for $u$, which in turn implies the Bernstein property, meaning that any positive solution to the minimal surface equation must be constant. As an application, we obtain oscillation estimates and a Bernstein Theorem for minimal graphs in products $M \times \mathbb{R}$, where $M$ is a smooth manifold (possibly weighted and with boundary) with non-negative Ricci curvature

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Theorems & Definitions (117)

• Definition
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Remark
• Theorem 5
• Theorem 6
• Definition 2.1
• Definition 2.2