Research in Classical Minimal Surface Theory

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

DMS-0405836 Title: Research in classical minimal surface theory PI: William H. Meeks, University of Massachusetts (Amherst) ABSTRACT Proposed Research Project Abstract In this proposal the researcher will study the geometry, asymptotic behavior, conformal structure and topology of properly embedded minimal surfaces in three-dimensional Euclidean space. One of the main goals of the proposal is to classify all of the properly embedded minimal surfaces which can be parametrized by domains in the Euclidean plane and to describe the asymptotic geometry of all finite genus examples. Related theoretical techniques concerning compactness, regularity and convergence of minimal surfaces of locally bounded genus will be investigated as well. One hoped for application of this research is to classify all smooth finite group actions on the three-dimensional sphere. As an outgrowth of his recent joint manuscript with Charles Frohman on the topological classification for minimal surfaces, the researcher proposes to prove that Bryant surfaces in hyperbolic three-space are unknotted. Classical minimal surface theory has its roots in 18-th and 19-th century mathematics. Minimal surfaces are the first important examples of what is called the calculus of variations, first described by Euler around 1735. Physically minimal surfaces can be modeled locally as soap films on wires or by surfaces of least-area relative to their local boundaries. Minimal surfaces play an important role as a tool in the study of three-dimensional topology and Riemannian geometry. The subject of minimal surfaces has a broad impact in mathematics and physical sciences. Minimal surfaces are stationary fluid interfaces,so their shapes arise in many physical problems. The work in this proposal would classify the possible physical shapes which could occur as infinite interfaces. Many of the known examples of minimal surfaces are observed physically, so it is of interest to have a rigorous theorem which predicts the shapes which can occur. The research proposed here strongly impacts the area of classical differential geometry of surfaces in three-dimensional Euclidean space. As is well known to geometers, minimal surface theory has been and continues to be one of the principal tools for proving theorems in general relativity and three-dimensional topology. One well-known such application is Schoen and Yau's proof of the Positive Mass Conjecture. Recent work of Gabai on the Generalized Smale Conjecture shows the continued importance of minimal surfaces in three-dimensional topology. Most of the research proposed here is related to and motivated by the hope that it will lead to a positive solution of the Pitts-Rubenstein Conjecture and to the classification of three-manifolds with finite fundamental group. This hoped for topological application to one of the outstanding classification problems in mathematics has its roots in previous joint research by the researcher, Peter Scott, Charles Frohman and S. T. Yau. In part because of the important connections with other areas of mathematics and the ease in which it is possible to make computer graphics pictures of beautiful classical examples, minimal surfaces continue to be one of the principal topics for popular science articles and public science exhibits. Thus, indirectly, the exciting research problems outlined in this proposal help bring many young scientists and mathematicians to the frontiers of research.