Research in Differential Geometry and Topology

摘要 提案：DMS-9803206主要研究者：William H.米克斯三世 本研究的总体目标是更深入地了解三维欧氏空间中曲面的整体几何。 一个目标是了解适当嵌入极小曲面的拓扑结构、渐近几何和共形结构。 第二个重要的目标是应用这些结果来了解收敛的有界亏格极小曲面在三流形。 极小曲面领域的起源是基于上世纪主要数学家的几何研究。 然而，新的分析和几何技术，在过去的二十年中发展的极小曲面理论在其他领域的一个重要的研究工具。 这些领域包括拓扑学、代数几何学、黑洞几何学和基本粒子物理学。 我建议如何描述这些无限曲面的几何行为，并对它们进行分类。 最近，这些表面的某些例子被用来模拟嵌段共聚物中的界面，我的一个分类结果表明，这些是唯一可能的例子。 这个建议的主要目标是证明关于这些曲面的一些美丽的命题，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.