Research in Differential Geometry and Topology

摘要提案：DMS-9803206 首席研究员：William H. Meeks III 本研究提案的总体目标是更深入地了解三维欧几里德空间中表面的全局几何形状。 目标之一是了解正确嵌入的最小曲面的拓扑、渐近几何和共形结构。 第二个重要目标是应用这些结果来理解三流形中有界亏格最小曲面的收敛性。 极小曲面领域的根源在于上个世纪主要数学家的几何研究。 然而，过去二十年来发展起来的新的分析和几何技术使得极小曲面理论成为其他领域的重要研究工具。 这些领域包括拓扑学、代数几何、黑洞几何和基本粒子物理学。 我建议描述这些无限表面的几何行为并对它们进行分类。 最近，这些表面的某些示例被用来模拟共嵌段聚合物中的界面，我的分类结果之一表明这些是唯一可能的示例。 该提案的主要目标是证明一些关于这些表面的美丽猜想，其解决方案将对数学和科学的其他部分有重要的应用。

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.