On parabolicity and area growth of minimal surfaces

TL;DR

This paper proves that certain minimal surfaces in R^3 exhibit parabolicity and quadratic area growth using stochastic methods, extending previous results without relying on superharmonic functions.

Contribution

It introduces a stochastic approach to establish parabolicity and area growth for minimal surfaces near cones, broadening the scope beyond catenoid-bound surfaces.

Findings

Minimal surfaces near cones are parabolic.

Such surfaces exhibit quadratic area growth.

Stochastic methods effectively analyze minimal surface properties.

Abstract

We establish parabolicity and quadratic area growth for minimal surfaces-with-boundary contained in regions of R^3 which are within a sub-logarithmic factor of the exterior of a cone. Unlike previous work showing that these two properties hold for minimal surfaces-with-boundary contained between two catenoids, we do not make use of universal superharmonic functions. Instead, we use stochastic methods, which have the additional feature of giving a type of parabolicity in a more general context than Brownian motion on a minimal surface.