Work

Minimal surface equation and Bernstein property on RCD spaces

Year: 2025

Type: article

Abstract: 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.

Source: Journal of Functional Analysis

DOI: 10.1016/j.jfa.2025.110907

Author: Alessandro Cucinotta

Institution: University of Oxford

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,250

Funder: Mathematical Institute, University of Oxford