Problems Related to Ricci Curvature

本文主要研究了以Ricci曲率为中心的几个问题。最近的发展显示了Ricci流的非凡力量。因此，它值得从不同的角度进行研究。首席研究员将研究Ricci流和比较几何，Ricci流的稳定性和紧Einstein流形。首席研究员还将研究非紧Einstein 4-流形的障碍和具有Ricci曲率下界的流形的Gromov-Hausdorff极限。流形等几何对象自然地出现在科学和工程中，作为配置空间，作为Einsten的宇宙模型。 Ricci曲率是几何学和爱因斯坦广义相对论中的一个基本概念。 爱因斯坦流形在数学和物理中都很重要。它们是广义黎曼流形上正则度规的良好候选者，也是广义相对论中爱因斯坦场方程（含宇宙常数）的真空解。广义相对论是研究引力的，引力是理解宇宙的一种基本力。因此，这些领域的基础研究不仅本身重要，而且在科学和工程方面也应该有意义。

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.