Problems Related to Ricci Curvature

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AbstractAward: DMS-0505733Principal Investigator: Guofang WeiThis proposal is concerned with several problems centered aroundRicci curvature. Recent development shows the extraordinary powerof Ricci flow. It thus deserves studies from various viewpoints.The principal investigator will study Ricci flow and comparisongeometry, stability of Ricci flow and compact Einstein manifolds.The principal investigator will also study obstructions fornoncompact Einstein 4-manifolds and the Gromov-Hausdorff limitsof manifolds with Ricci curvature lower bound.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 geometryas well as Einstein's general relativity. Einstein manifolds areimportant both in mathematics and physics. They are goodcandidates for canonical metrics on general Riemannian manifoldsand they are the vacuum solutions of Einstein's field equation(with cosmological constant) in general relativity. Generalrelativity is the study of gravity, the one fundamental forcethat should play a crucial role in understanding ouruniverse. Thus the fundamental research in these areas should notonly be important in its own right but also should haveimplications in science and engineering.