Parabolic flows in geometry

PI建议研究几何偏微分方程组领域的各种问题。例如，他想研究具有正各向同性曲率的流形上的Ricci流的定性行为。特别是，PI想要分析沿流动可能发生的奇点类型。在另一个项目中，PI打算研究带边界流形上的Yamabe问题。这里的目标是构造内部具有恒定标量曲率且沿边界平均曲率为零的共形度量。PI还打算研究n维球面上Yamabe流的古代解。这个问题的动机是由于汉密尔顿、达斯卡洛普洛斯和塞苏姆对二维球面上利玛窦流动的古代解的分类结果。这个项目是关于分析和微分几何的十字路口的问题。近几年来，几何学中分析技术的使用极为成功。特别是，在Perelman解Poincare猜想以及Richard Schoen和我对可微球定理的证明中，Hamilton的Ricci流起到了关键作用。该项目的目标是更好地了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.