Minimal Surfaces of Least Total Curvature and Moduli Spaces of Plane Polygonal Arcs

Abstract

Introduction. The first major goal of this paper is to prove the existence of complete minimal surfaces of each genus p > 1 which minimize the total curvature (equivalently, the degree of the Gaus map) for their genus. The genus zero version of these surfaces is known as Enneper’s surface (see [Oss2]) and the genus one version is due to Chen-Gackstatter ([CG]). Recently, experimental evidence for the existence of these surfaces for genus p ≤ 35 was found by Thayer ([Tha]); his surfaces, like those in this paper, are hyperelliptic surfaces with a single end, which is asymptotic to the end of Enneper’s surface. Our methods for constructing these surfaces are somewhat novel, and as their development is the second major goal of this paper, we sketch them quickly here. As in the construction of other recent examples of complete immersed (or even embedded) minimal surfaces in E, our strategy centers around the Weierstras representation for minimal surfaces in space, which gives a parametrization of the minimal surface in terms of meromorphic data on the Riemann surface which determine three meromorphic one-forms on the underlying Riemann surface. The art in finding a minimal surface via this representation lies in finding a Riemann surface and meromorphic data on that surface so that the representation is well-defined, i.e., the local Weierstras representation can be continued around closed curves without changing its definition. This latter condition amounts to a condition on the imaginary parts of some periods of forms associated to the original Weierstras data. In many of the recent constructions of complete minimal surfaces, the geometry of the desired surface is used to set up a space of possible Weierstras data and Riemann surfaces,