Curvature rigidity, quasi-local mass and related problems

摘要：研究人员：王晓东（音）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。本课题将研究几个涉及标量曲率的此类问题。首先是了解具有凸边界和正标量曲率的流形的边界效应。这与广义相对论中对准局部质量的理解密切相关。一个关键的特殊情况是，一个标量曲率大于或等于6的紧致三流形，其边界是完全测地线且与标准两球等距的，是否与三维半球等距。第二个问题是紧化双曲三流形上的Yamabe数是否由双曲度规得到。如果这一结论成立，将是黎曼几何若干深刻结果的显著推广。它等价于双曲度规是否比具有相同曲率的其他度规体积小的问题。这些问题肯定会成为灵感的巨大来源，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增强我们对时空结构的理解。同样可以想象的是，理解不同曲率假设下的边界效应将在其他科学和工程领域得到应用。希望本课题对本科生和研究生的教育与培养也能起到积极的作用。这将是作者与他的同事们共同努力的一部分，以加强密歇根州立大学几何分析研究生课程。

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.