Some Rigidity and Comparison Problems Involving the Scalar or Ricci Curvature

微分几何的中心主题之一是理解曲率及其在几何和拓扑性质方面的含义。本课题研究涉及标量曲率和里奇曲率的两类问题。第一类问题涉及边界为标量或里奇曲率为正的紧致黎曼流形。如何理解这些曲率条件的边界效应是一个重要的问题。这些问题与理解广义相对论中的准局部质量有关。一个关键的特殊情况是标量曲率大于或等于6的紧致3流形，其边界完全是测地线且与标准二维球体等距，是否与三维半球等距。这些问题也将成为灵感的重要来源，并导致许多其他有趣的问题。具有非负里奇曲率的黎曼流形已经被研究了很多，我们对它们有了很好的了解。里奇曲率负下界的黎曼流形更复杂，也更不容易理解。作者打算通过研究一些涉及熵和拉普拉斯算子谱等渐近不变量的刚性和比较问题来研究它们。本项目旨在研究涉及标量和里奇曲率的一些基本问题。进步将加深我们对曲率和几何的理解。这些问题与数学的其他领域有相互作用，包括代数几何、概率论和势理论。其中一些问题与理论物理学密切相关，特别是广义相对论，它们的解决方案将增强我们对时空的理解。该项目还将有助于培养几何分析领域的研究生和博士后。

One of the central themes in differential geometry is to understand curvature and its implications in terms of geometric and topological properties. This project is to study two classes of problems involving scalar curvature and Ricci curvature. The first class of problems concerns compact Riemannian manifolds with boundary whose scalar or Ricci curvature is positive. It is an important problem to understand the boundary effect of these curvature conditions. These problems are related to understanding quasi-local mass in general relativity. A key specific case is whether a compact 3-manifold with scalar curvature bigger or equal to six whose boundary is totally geodesic and isometric to the standard two dimensional sphere is isometric to the three dimensional hemisphere. These problems will also serve as great source of inspiration and lead to many other fascinating problems. Riemannian manifolds with nonnegative Ricci curvature have been studied a lot and we have a good knowledge about them. Riemannian manifolds with a negative lower bound for Ricci curvature are more complicated and less understood. The author intends to study them by working on some rigidity and comparison problems involving asymptotic invariants such as entropy and the spectrum of the Laplace operator. This project aims to study some fundamental problems involving scalar and Ricci curvature. Progress will deepen our understanding of curvature and geometry. These problems have interactions with other areas of mathematics including algebraic geometry, probability and potential theory. Some of these problems are closely related to theoretical physics, particularly general relativity and their solutions will enhance our understanding of spacetime. This project will also contribute to the training of graduate students and post-docs in the area of geometric analysis.