Curvature rigidity, quasi-local mass and related problems

AbstractAward：DMS-0505645首席研究员：王晓东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。尽管在过去的几十年里取得了巨大的进步，但仍然存在许多根本性的问题。 本课题就是研究这类涉及标量曲率的问题。第一部分是了解具有凸边界和正数量曲率的流形上的边界效应。这与广义相对论中对准定域质量的理解密切相关。一个关键的特例是一个标量曲率大于或等于6的紧致三维流形，其边界是全测地的且与标准二球面等距，是否与三维半球等距.第二个问题是紧双曲三流形上的Yamabe数是否由双曲度量得到。如果是真的，这将是黎曼几何中几个深层次结果的显著推广.这等价于双曲度量是否比任何曲率相同的度量具有更小的体积的问题。这些问题无疑将成为一个巨大的灵感来源，并导致许多其他迷人的问题，这个项目将产生广泛的影响以外的领域的几何分析.这些重要的问题与理论物理，特别是广义相对论密切相关。它们的解将极大地增强我们对时空结构的理解。 也可以想象，在各种曲率假设下理解边界效应将在科学和工程的其他领域中有应用。笔者希望本项目也能对本科生和研究生的教育与培训做出积极的贡献。 这将是作者与同事们共同努力加强密歇根州立大学几何分析研究生课程的一部分。

AbstractAward: DMS-0505645Principal Investigator: Xiaodong WangOne of the central themes in differential geometry is tounderstand curvature and its implications in terms of geometricand topological properties. Despite the enormous progress madeduring the last several decades, there remain many fundamentalproblems. This project is to study several such problemsinvolving the scalar curvature. The first one is to understandthe boundary effect on manifolds with convex boundary andpositive scalar curvature. This is closely related tounderstanding quasi-local mass in general relativity. A keyspecific case is whether a compact three-manifold with scalarcurvature bigger or equal to six whose boundary is totallygeodesic and isometric to the standard two-sphere is isometric tothe three dimensional hemisphere. The second problem is whetheron a compact hyperbolic three-manifold the Yamabe number isachieved by the hyperbolic metric. If true it will be aremarkable generalization of several deep results in Riemanniangeometry. It is equivalent to the question whether the hyperbolicmetric has smaller volume than any other metric with the samescalar curvature. These problems definitely will serve as greatsource of inspiration and lead to many other fascinatingproblems.This project will have a broad impact outside the field ofgeometric analysis. These important problems are closely relatedto theoretical physics, particularly general relativity. Theirsolutions will greatly enhance our understanding of spacetimestructures. It is also conceivable that understanding boundaryeffect under various curvature assumptions will have applicationsin other areas of science and engineering. The author hopes thatthe project will also have a positive contribution toundergraduate and graduate education and training. It will bepart of the author's joint efforts with his colleagues tostrengthen the graduqate program in geometric analysis atMichigan State University.