Minimal 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 embedded minimal surfaces and possible applications of these results to basic research in three-dimensional topology and geometry. 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. As an outgrowth of his recent joint manuscript with Charles Froman 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 boundaries. Minimal surfaces represent stationary fluid interfaces, and so their shapes arise in many physical problems. The work in this proposal will help classify the possible physical shapes which might occur as such interfaces. Many of the known examples of minimal surfaces are observed physically, and so it is of interest to have a rigorous theorem which predicts the shapes which can occur. In part because of important connections with other areas of mathematics and because it is possible to make beautiful computer graphics pictures of classical examples, minimal surfaces continue to be one of the principal topics for popular science articles and public science exhibits. Thus, indirerctly, the exciting research problems outlined in this proposal help bring many young scientists and mathematicians to the froniters of research.
