Aspects of Bakry-Emery Ricci Curvature

AbstractAward：DMS-1105536主要研究员：魏国芳本论文研究了具有Bakry-Emery Ricci曲率下有界的光滑度量测度空间和拟爱因斯坦度量的几何和拓扑. Bakry-Emery Ricci曲率是Ricci曲率在光滑度量测度空间上的重要推广，它自然地作为塌陷的测度Gromov-Haudorff极限而出现.此外，Bakry-Emery Ricci曲率在Ricci孤子、弯曲爱因斯坦度量、扩散过程、对数Sobolev不等式的研究以及度量测度空间Ricci曲率下界的推广等方面都有着重要的作用.主要研究了Ricci孤子的几何、比较几何与Ricciflow的关系以及Ricci正、非负下界的第一特征值的联合估计。 PI还将研究具有较低积分Ricci曲率界的流形以及具有极小体积的双曲轨道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，从而导致了Poincare猜想的解决. 准爱因斯坦度量在数学和物理学中都有重要的意义。 光滑度量测度空间也与最优输运、信息几何、离散几何等有关。拟议的活动将对所有这些方面产生影响。 基本群是最基本的（顾名思义）拓扑信息.对它的理解将极大地推进曲率界对黎曼流形整体拓扑的影响的研究，从而有助于回答宇宙的形状问题。

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.