Existence and Geometry of Complete Riemannian Structures

9704515波拉克这个项目位于黎曼几何学领域。更具体地说，研究人员将在维度大于2的流形上研究具有常正标量曲率的完备共形平坦度量；他还将构造嵌入欧氏空间的常平均曲率曲面的完备曲面(CMC曲面)。物理学家经常试图将宇宙描述为具有某种特殊几何类型的黎曼流形。常正标量曲率的条件就是这种几何的一个例子。流形上的共形平坦结构是给定角度的曲面上结构的高维推广。CMC曲面的示例包括肥皂泡。这些表面试图最小化其表面积，同时保持固定体积的空气封闭。

9704515 Pollack This project lies in the area of Riemannian geometry. More specifically, the investigator will examine complete conformally flat metrics of constant positive scalar curvature on manifolds of dimension greater than two; he will also construct complete surfaces of constant mean curvature surfaces embedded in Euclidean space (CMC surfaces). Physicists often seek to describe the universe as a Riemannian manifold possessing a certain special type of geometry. The condition of constant positive scalar curvature is an example of such geometry. A conformally flat structure on a manifold is a generalization to higher dimensions of the structure on a surface in which angles are prescribed. Examples of CMC surfaces include soap bubbles. These are surfaces which try to minimize their surface area while keeping a fixed volume of air enclosed.