Research in Differential Geometry and Topology

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

Abstract for DMS - 0104044 In this proposal the research will study the geometry, asymptotic behavior, conformal structure and topology of properly embedded minimal surfaces in R3 . One of the main goals of the proposal is to classify all properly embedded minimal surfaces of genus zero 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 S3 . Finally the research proposes to prove that Bryant surfaces in hyperbolic three-space are unknotted. Classical minimal surface theory has its roots in 18th and 19th century mathematics and gives one of the first important examples in what is called the calculus of variations, first described by Euler. Physically minimal surfaces can be modeled locally as soap films on wires or by surfaces of least-area relative to local boundaries. These surfaces play an important role as a tool in the study of three-dimensional topology and Riemannian geometry. The research in this proposal concerns global properties of these surfaces and possible applications to basic research in three-dimensional topology and geometry.