Minimal Surface Equation and Bernstein Property on RCD spaces

TL;DR

This paper extends the Bernstein property to solutions of the minimal surface equation on RCD spaces, showing that positive solutions are constant and deriving related oscillation estimates and Bernstein theorems.

Contribution

It proves that solutions to the minimal surface equation on RCD(K,N) spaces are harmonic on their graphs and establishes the Bernstein property in this setting.

Findings

Solutions are harmonic on their graphs in RCD spaces.

Positive solutions to the minimal surface equation are constant when K=0.

Derived oscillation estimates and Bernstein theorems for minimal graphs in product spaces.

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