Existence and Geometry of Complete Riemannian Structures

点击翻译按钮获取中文摘要

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.