Curvature rigidity, quasi-local mass and related problems

Michigan State University, East Lansing MI

Investigators

Abstract

Abstract Award: DMS-0505645 Principal Investigator: Xiaodong Wang One of the central themes in differential geometry is to understand curvature and its implications in terms of geometric and topological properties. Despite the enormous progress made during the last several decades, there remain many fundamental problems. This project is to study several such problems involving the scalar curvature. The first one is to understand the boundary effect on manifolds with convex boundary and positive scalar curvature. This is closely related to understanding quasi-local mass in general relativity. A key specific case is whether a compact three-manifold with scalar curvature bigger or equal to six whose boundary is totally geodesic and isometric to the standard two-sphere is isometric to the three dimensional hemisphere. The second problem is whether on a compact hyperbolic three-manifold the Yamabe number is achieved by the hyperbolic metric. If true it will be a remarkable generalization of several deep results in Riemannian geometry. It is equivalent to the question whether the hyperbolic metric has smaller volume than any other metric with the same scalar curvature. These problems definitely will serve as great source of inspiration and lead to many other fascinating problems. This project will have a broad impact outside the field of geometric analysis. These important problems are closely related to theoretical physics, particularly general relativity. Their solutions will greatly enhance our understanding of spacetime structures. It is also conceivable that understanding boundary effect under various curvature assumptions will have applications in other areas of science and engineering. The author hopes that the project will also have a positive contribution to undergraduate and graduate education and training. It will be part of the author's joint efforts with his colleagues to strengthen the graduqate program in geometric analysis at Michigan State University.