Minimal surface equation and Bernstein property on RCD spaces

Abstract

We show that if $(X,d,m)$ is an $\mathrm{RCD}(K,N)$ space and $u\in W_{loc}^{1,1}\left(X\right)$ 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 $M\times R$, where $M$ is a smooth manifold (possibly weighted and with boundary) with non-negative Ricci curvature.