Parabolic flows in geometry

PI建议研究几何偏微分方程领域的各种问题。例如，他想研究的定性行为的里奇流流形与积极的各向同性曲率。特别地，PI想要分析什么样的奇点可以沿流沿着发生。在另一个项目中，PI打算研究具有边界的流形上的Yamabe问题。这里的目标是构造共形度量，具有恒定的标量曲率的内部和零平均曲率沿着边界。 PI还打算研究古代的解决方案Yamabe流的n维球。这个问题的动机是一个分类的结果，由于汉密尔顿，Daskalopoulos，和Sesum，古代解决方案的里奇流的两个领域。这个项目关注的是在分析和微分几何的十字路口的问题。近年来，几何学中分析技巧的应用极为成功。特别是，汉密尔顿的里奇流起着关键作用，佩雷尔曼的解决方案的庞加莱猜想，以及在证明微分球定理的理查德舍恩和我自己。该项目的目标是更好地理解这些几何演化方程及其定性性质。

The PI proposes to study various problems in the field of geometric partial differential equations. For example, he would like to study the qualitative behavior of the Ricci flow on manifolds with positive isotropic curvature. In particular, the PI would like to analyze what kinds of singularities can occur along the flow. In another project, the PI intends to study the Yamabe problem on manifolds with boundary. The goal here is to construct conformal metrics that have constant scalar curvature in the interior and zero mean curvature along the boundary. The PI also intendsto study ancient solutions to the Yamabe flow on the n-dimensional sphere. This question is motivated by a classification result, due to Hamilton, Daskalopoulos, and Sesum, for ancient solutions to the Ricci flow on the two-sphere. This project is concerned with questions at the crossroads of analysis and differential geometry. The use of analytical techniques in geometry has been extremely successful in recent years. In particular, Hamilton's Ricci flow plays a key role in Perelman's solution of the Poincare conjecture, as well as in the proof of the Differentiable Sphere Theorem by Richard Schoen and myself. The goal of the project is to gain a better understanding of these geometric evolution equations, and their qualitative properties.