Aspects of Bakry-Emery Ricci Curvature

摘要奖：DMS-1105536首席研究员：魏国芳这项建议研究了具有Bakry-Emery Ricci曲率下界的光滑度量空间和拟爱因斯坦度量空间的几何和拓扑。Bakry-Emery RiccicCurvature是Ricci曲率在光滑度量空间中的一个重要推广，它是作为可测Gromov-Haudorff极限的折叠而自然产生的。此外，Bakry-Emery Ricci曲率在研究Ricci孤子、翘曲爱因斯坦度量、扩散过程、对数Sobolov不等式以及推广度量空间的Ricci曲率下界等问题中起着重要的作用。主要研究人员将研究Ricci孤子的几何，比较几何和Ricciflow之间的关系，并得到Ricci正下界和非负下界的第一本征值的组合估计。PI还将研究具有较低积分Ricci曲率界和最小双曲奥氏体体积的流形的基本群的结构。Bakry-Emery Ricci曲率和Ricci孤子在Ricci流中起着非常重要的作用，这导致了Poincare猜想的解。准爱因斯坦度规在数学和物理中都很重要。光滑度量空间还与最优传输、信息几何、离散几何有关。拟议的活动将对所有这些方向产生影响。基本群是最基本(顾名思义)的拓扑信息。它的理解将极大地促进曲率界对黎曼流形全局拓扑影响的研究，从而有助于回答有关宇宙形状的问题。

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.