The Topology, Geometry and Conformal Structure of Properly Embedded Minimal Surfaces

Abstract

This paper develops new tools for understanding surfaces with more than one end and infinite topology which are properly minimally embedded in Euclidean three-space. On such a surface, the set of ends forms a totally disconnected compact Hausdorff space, naturally ordered by the relative heights of the ends in space. One of our main results is that the middle ends of the surface have quadratic area growth, and are thus not limit ends. This implies that the surface can have at most two limit ends, which occur at the top and bottom of the ordering, and thus only a countable number of ends, which is a strong topological restriction. There are also restrictions on the asymptotic geometry and conformal structure of such a surface: for example, we prove that if the surface has exactly two limit ends, then it is recurrent (that is, almost all Brownian paths are dense in the surface), and in particular, any positive harmonic function on the surface is constant. These results have played an important role in several recent advances in the theory, including the uniqueness of the helicoid, the invariance of flux for a coordinate function on a properly immersed minimal surface, and the topological classification of properly embedded minimal surfaces of finite genus.