Mathematical Sciences: Research in Differential Geometry and Topology

9505101 Meek提出了极小和常平均曲率曲面的微分几何和拓扑中的各种问题。对于有限全曲率的完备极小曲面，将寻求全高斯曲率结果以及拓扑阻碍结果。对于无限全曲率极小曲面，我们将研究拓扑唯一性、增长估计和指数理论。我们将研究常平均曲率曲面的整体性质。还将对其中一些曲面进行计算机图形学和(周期)计算。比利时物理学家Platform在他著名的肥皂膜实验中首次系统地研究了极小曲面和常平均曲率曲面。这些曲面对物理曲面进行建模，其中表面面积通过表面张力最小化，或者在两种材质之间具有稳定界面的情况下。近年来，周期极小曲面被用来在分子尺度上模拟凝聚态物质。

9505101 Meeks A variety of problems in differential geometry and topology of minimal and constant mean curvature surfaces are proposed. Total Gaussian curvature results as well as topological obstruction results will be sought for complete minimal surfaces of finite total curvature. For infinite total curvature minimal surfaces, topological uniqueness, growth estimates and the index theory will be examined. Global properties of constant mean curvature surfaces will be studied. Computer graphics and (period) computations for some of these surfaces will also be pursued. Minimal surfaces and constant mean curvature surfaces were first systematically investigated by the Belgian physicist Plateau in his famous soap film experiments. These surfaces model physical surfaces where either the surface area is minimized by surface tension or in the case of a stable interface between two materials. In recent years, periodic minimal surfaces are used to model condensed matter on a molecular scale.