Curvature rigidity, quasi-local mass and related problems

摘要奖：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。尽管在过去几十年中取得了巨大进展，但仍然存在许多根本性问题。本课题研究涉及标量曲率的几个这样的问题。第一个是理解具有凸边界和正数量曲率的流形上的边界效应。这与理解广义相对论中的准局域质量密切相关。一个关键的特殊情况是一个紧致的三维流形，其标量曲率大于或等于6，其边界是全测地的，并且等距于标准的两个球面，是否等距于三维半球。第二个问题是紧致双曲三流形上的Yamabe数是否由双曲度规获得。如果是真的，这将是黎曼计量学中几个深层次结果的显著推广。它等价于双曲度量是否比具有角曲率的其他度量具有更小的体积的问题。这些问题肯定会成为灵感的巨大源泉，并导致许多其他有趣的问题。这个项目将在几何分析领域之外产生广泛的影响。这些重要问题与理论物理，特别是广义相对论密切相关。他们的解决方案将极大地提高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得到应用。希望该项目对研究生和研究生教育培训工作也能起到积极的推动作用。这将是作者与他的同事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.