The asymptotic behavior of properly embedded minimal surfaces of finite topology

Abstract

While advances have been made in recent years in the study of properly embedded minimal surfaces of finite topology in R3 [3, 5, 8, 9, 11, 12, 13, 23], progress has depended, in an essential manner, on the special structure of such surfaces with finite total curvature. A properly immersed minimal surface with finite total curvature is, conformally, a compact Riemann surface punctured in a finite number of points, and its Gauss map extends to the compact surface as a meromorphic function [22]. In particular, all the topological ends are conformally equivalent to a punctured disk, and there is a well-defined limit tangent plane at each end. Outside of a sufficiently large compact set, such an end is a multisheeted graph over the limit tangent plane, and if the end is embedded, it is asymptotic to either a plane or a half-catenoid. (See, e.g., Schoen [24]. For a survey of results on properly immersed surfaces of finite total curvature, see [9, ? 1].) Without the assumption of finite total curvature, these results are not true in general. As an indication of the relative lack of information, consider that it is unknown whether or not the helicoid is the only simply-connected minimal surface that is properly embedded and nonplanar (a question raised by Osserman). The helicoid has infinite total curvature and a single annular end.' If a surface has finite topology, then all of its ends are annular. Any general theory of properly embedded minimal surfaces of finite topology must include an understanding of the behavior of these ends. In this paper, we prove