Some Rigidity and Comparison Problems Involving the Scalar or Ricci Curvature

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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.