Mathematical Sciences: Research in Differential Geometry andTopology

首席研究员将研究各种问题， 微分几何与微分拓扑 的主要目标 这项研究旨在加深对全球 R3中曲面的拓扑和几何性质， 满足一些几何或变分原理。 特别 重点是：1. 全局几何和拓扑 适当嵌入极小曲面：存在定理， 拓扑障碍，全曲率定理，共形 结构，拓扑唯一性，指数计算 例子，等距的扩展，最小图的唯一性， 非紧域上的对数增长，面积增长估计 目的 2. R3中常平均曲率的整体几何， 双曲三维空间 3. 计算有关的问题， 话题吧 4. R3中的合痕曲面几何。 几何学中的一个经典话题是曲面的性质， 具有特定边界的最小区域，即所谓的肥皂膜 问题，这可以通过将线框浸入 肥皂溶液中，并检查其撤回时形成的膜。 忽略重力，已知这种薄膜具有高斯曲率 到处都是零。 一个稍微更一般的物理情况要求 对于曲率恒定但不一定为零的曲面。 关于这种表面的问题比比皆是，虽然很自然，而且经常 简单地说，他们是令人惊讶的困难，并启发了 创建强大的方法，借鉴不同领域的 数学 这样的问题和方法是这其中的核心 项目

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problem, which can be illustrated by dipping a wire frame into a soap solution and examining the film formed when it is withdrawn.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.