Minimal surface equation and Bernstein property on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="sans-serif">RCD</mml:mi></mml:math> spaces

We show that if ( X , d , m ) is an RCD ( K , N ) space and u ∈ W l o c 1 , 1 ( 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 × R , where M is a smooth manifold (possibly weighted and with boundary) with non-negative Ricci curvature.