Abstract Proposal: DMS-9803206 Principal Investigator: William H. Meeks III The general goal of this research proposal is to get a deeper understanding of the global geometry of surfaces in three-dimensional Euclidean space. One goal is to understand the topology, asymptotic geometry and conformal structure of properly embedded minimal surfaces. A second important goal is to apply these results to understand the convergence of bounded genus minimal surfaces in three-manifolds. The field of minimal surfaces has its roots based on geometric studies by the major mathematicians of the previous century. However, the new analytic and geometric techniques developed in the past twenty years have made the theory of minimal surfaces an essential research tool in other fields. These fields include Topology, Algebraic Geometry, geometry of black holes and elementary particle Physics. I propose to characterize how these infinite surfaces behave geometrically and to classify them. Recently certain examples of these surfaces were used to model interfaces in coblock polymers and one of my classification results shows that these are the only possible examples. The primary goal of this proposal is to prove some beautiful conjectures concerning these surfaces, whose solutions will have important applications to other parts of Mathematics and Science.
