Curvature rigidity, quasi-local mass and related problems

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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.