Manifolds with Lower Ricci Curvature Bounds

摘要-DMS-0806016这项建议是研究具有点态或积分Ricci曲率下界的流形。最近的发展表明了利玛窦流动的非凡力量。因此，它值得从多个角度进行研究。PI将通过比较几何来研究Ricci流。许多几何问题导致积分曲率，例如等谱问题、几何变分问题和极值度量，以及特征数的Chern-Weil公式。因此，积分曲率界可视为最优曲率假设。基本群是最基本的拓扑信息。PI将研究具有下积分Ricci曲率界的流形的基本群的结构。它的理解将极大地促进对曲率界对黎曼流形全局拓扑影响的理解。Ricci曲率是几何中的一个基本概念，也是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很好候选者，它们是广义相对论中爱因斯坦场方程式(具有宇宙常数)的真空解。广义相对论是对引力的研究，引力是理解我们的宇宙应该发挥关键作用的一个基本力。拟议的活动将对所有这些方向产生影响。

Abstract-DMS-0806016This proposal is to study manifolds with lower point-wise or integral Ricci curvature bounds. Recent development shows the extraordinary power of Ricci flow. It thus deserves studies from various viewpoints. The PI will study Ricci flow via comparison geometry. Many geometric problems lead to integral curvatures; for example, the isospectral problems, geometric variational problems and extremal metrics, and Chern-Weil's formula for characteristic numbers. Thus, integral curvature bounds can be viewed as an optimal curvature assumption. The fundamental group is the most basic topological information. The PI will study the structures of the fundamental groups for manifolds with lower integral Ricci curvature bound. Its understanding will greatly advance the understanding of the effect of curvature bounds on the global topology of Riemannian manifolds.Ricci curvature is a fundamental concept in geometry as well as Einstein's general relativity. Einstein manifolds are important both in mathematics and physics. They are good candidates for canonical metrics on general Riemannian manifolds and they are the vacuum solutions of Einstein's field equation (with cosmological constant) in general relativity. General relativity is the study of gravity, the one fundamental force that should play a crucial role in understanding our universe. The proposed activities would have impact on all these directions.