A martingale approach to minimal surfaces

TL;DR

This paper introduces a probabilistic framework using Brownian motion to analyze minimal surfaces in three-dimensional space, providing new insights into their geometric properties and related theorems.

Contribution

It presents a novel coupling of Brownian motions on minimal surfaces to derive maximum principle results and Liouville theorems.

Findings

Coupling of Brownian motions on minimal surfaces

Proofs of maximum principle-type results

New Liouville theorems for minimal surfaces

Abstract

We provide a probabilistic approach to studying minimal surfaces in three-dimensional Euclidean space. Following a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used to study two classes of results in the theory of minimal surfaces, maximum principle-type results, such as weak and strong halfspace theorems and the maximum principle at infinity, and Liouville theorems.