A martingale approach to 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.

33 pages, exposition in Section 3 re-worked, minor corrections, one reference added

Cited by in corpus (2)

• Brownian motion and the parabolicity of minimal graphs
• Stochastic half-space theorems for minimal surfaces and $H$-surfaces of ${\mathbb{R}}^3$