Mathematical Sciences: Research in Differential Geometry andTopology

首席研究员将研究微分几何和微分拓扑中的各种问题。该研究的主要目标是对满足一些好的几何或变分原理的R3曲面的全局拓扑和几何性质有更深入的了解。将特别强调：1。适当嵌入最小曲面的整体几何和拓扑：存在性定理、拓扑障碍、总曲率定理、共形结构、拓扑唯一性、实例索引的计算、等距的扩展、非紧域上对数增长最小图的唯一性、端点的面积增长估计。2. 在R3和双曲三维空间中恒定平均曲率的全局几何。3. 与本学科问题相关的计算。4. R3中同位素表面的几何形状。几何中的一个经典主题是具有特定边界的最小面积表面的性质，即所谓的肥皂膜问题，可以通过将金属丝框架浸入肥皂溶液中并检查其取出时形成的薄膜来说明。忽略重力，这样的薄膜到处都有零高斯曲率。稍微更一般的物理情况要求曲面曲率恒定但不一定为零。关于这类曲面的问题很多，虽然很自然，而且往往很简单，但它们却出奇地困难，并激发了利用数学各个领域的强大方法的创造。这些问题和方法是这个项目的核心。

The principal investigator will study a variety of problems in differential geometry and differential topology. The main goal of the research is to develop a deeper understanding of global topological and geometrical properties of surfaces in R3 that satisfy some nice geometrical or variational principle. Particular emphasis will be placed upon: 1. The global geometry and topology of properly embedded minimal surfaces: existence theorems, topological obstructions, total curvature theorems, conformal structure, topological uniqueness, computation of the index of examples, extension of isometries, uniqueness of minimal graphs of logarithmic growth over noncompact domains, area growth estimates for ends. 2. Global geometry of constant mean curvature in R3 and hyperbolic 3-space. 3. Computation relevant to problems in the subject. 4. Geometry of isotopy surfaces in R3. A classical topic in geometry is the nature of surfaces of minimal area having a specified boundary, the so-called soap film problem, which can be illustrated by dipping a wire frame into a soap solution and examining the film formed when it is withdrawn. Ignoring gravity, such films are known to have Gaussian curvature zero everywhere. A slightly more general physical situation calls for surfaces with constant but not necessarily zero curvature. Problems about such surfaces abound, and while natural and often simple to state, they are surprisingly difficult and have inspired the creation of powerful methods drawing on diverse areas of mathematics. Such problems and methods are the heart of this project.