Geometry, Analysis and Topology under Curvature Bounds

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AbstractAward: DMS-9971833Principal Investigator: Guofang WeiThis proposal is concerned with the effect of Ricci curvature andintegral curvature bounds on the local geometric structure andglobal topology of a Riemannian manifold.The principal investigator will study (i) deformation of metrics to positive sectional curvature, (ii) optimal bound of isoperimetric constant, (iii) the structures of the fundamental groups for manifolds with pointwise or integral lower Ricci curvature bound, and(iv) structures of manifolds with bounded weak norm. Geometric objects such as manifolds appear naturally in scienceand engineering, as configuration spaces, as Einsten's model ofuniverse. Ricci curvature is a fundamental concept in Einstein'sgeneral relativity. Thus the fundamental research in these areashould not only be important in its own right but also shouldhave implications in science and engineering.