A martingale approach to minimal surfaces

Details A martingale approach to minimal surfaces A martingale approach to minimal surfaces

2008-05-05

arxiv.org/abs/0805.0556v2

J. Funct. Anal. 256 (2009), no. 8, 2440-2472

Mathematics

Differential Geometry

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

Scientific paper

10.1016/j.jfa.2008.06.033

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.

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