Mathematical Sciences: Research in Differential Geometry andTopology

主要研究人员将研究微分几何和微分拓扑学中的各种问题。研究的主要目的是加深对R3中满足某些几何或变分原理的曲面的全局拓扑和几何性质的理解。1.适当嵌入极小曲面的整体几何和拓扑：存在定理、拓扑障碍、全曲率定理、共形结构、拓扑唯一性、实例指数的计算、等距扩张、非紧域上对数增长极小图的唯一性、端部的面积增长估计。2.R3中常平均曲率的整体几何和三维双曲空间。3.与本学科问题相关的计算。4.R3中等距曲面的几何。几何中的一个经典主题是具有指定边界的最小面积曲面的性质，即所谓的肥皂膜问题，可以通过将线框浸入肥皂溶液中并检查当它被取出时形成的膜来说明。忽略重力，这样的电影到处都是高斯曲率为零的。稍微更一般的物理情况需要具有恒定但不一定为零曲率的曲面。关于这类曲面的问题比比皆是，虽然这很自然，而且往往很容易描述，但它们出人意料地困难，并启发了利用不同数学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problem, which can be illustrated by dipping a wire frame into a soap solution and examining the film formed when it is withdrawn.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.