A stochastic approach to parabolicity and area growth of minimal ends

Stochastics Seminar
Thursday, February 23, 2012 - 3:05pm
1 hour (actually 50 minutes)
Lehigh University
We wish to understand ends of minimal surfaces contained in certain subsets of R^3. In particular, after explaining how the parabolicity and area growth of such minimal ends have been previously studied using universal superharmonic functions, we describe an alternative approach, yielding stronger results, based on studying Brownian motion on the surface. It turns out that the basic results also apply to a larger class of martingales than Brownian motion on a minimal surface, which both sheds light on the underlying geometry and potentially allows applications to other problems.