Aspects of Bakry-Emery Ricci Curvature

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AbstractAward: DMS-1105536Principal Investigator: Guofang Wei This proposal studies the geometry and topology of smooth metricmeasure spaces with Bakry-Emery Ricci curvature bounded frombelow and quasi-Einstein metrics. The Bakry-Emery Riccicurvature is an important generalization of Ricci curvature forsmooth metric measure space, which occurs naturally as thecollapsed measured Gromov-Haudorff limit. Furthermore, theBakry-Emery Ricci curvature plays important roles in a variety oftopics such as the study of Ricci soliton, warped Einsteinmetric, diffusion process, logrithmic Sobolev inequality, as wellas in the extension of Ricci curvature lower bound for metricmeasure space. The principal investigator will study the geometryof Ricci solitons; relation between comparison geometry and Ricciflow and obtaining a combined estimate for the first eigenvaluefor Ricci positive and nonnegative lower bound. The PI will alsostudy the structures of the fundamental groups for manifolds withlower integral Ricci curvature bound as well as minimal volume ofhyperbolic orbifolds.Bakry-Emery Ricci curvature and Ricci soliton play a veryimportant role in Ricci flow, which led to the solution ofPoincare conjecture. Quasi-Einstein metrics is are importantboth in mathematics and physics. Smooth metric measure space isalso related to optimal transport, information geometry, discretegeometry. The proposed activities would have impact on all thesedirections. The fundamental group is the most fundamental (asthe name implies) topological information. Its understanding willgreatly advance the study of the effect of curvature bounds onthe global topology of Riemannian manifolds, therefore will behelpful in answering questions about the shape of the universe.