Mean curvature flow and Ricci flow

这项研究涉及到了微分几何、偏微分方程组、变分法、随机分析和广义相对论。具体地说，主要集中在热方程的两个几何版本上：通过曲面的平均曲率来演化曲面，以及通过汉密尔顿的Ricci流来演化曲面空间。平均曲率流模拟了许多涉及不断变化的表面或界面的物理过程。这是减少曲面面积并使其向最佳曲面演化的最有效方法。相应地，Ricci流将曲线空间变形为最佳形状。

The proposed research is at the intersection of differential geometry, partial differential equations, calculus of variations, stochastic analysis and general relativity. Specifically, the main focus is on two geometric versions of the heat equation: the evolution of surfaces by their mean curvature, and the evolution of curved spaces by Hamilton’s Ricci flow. Mean curvature flow models many physical processes which involve an evolving surface, or interface. It is the most efficient way to decrease the area of surfaces and to evolve them towards optimal ones. Correspondingly, Ricci flow deforms curved spaces towards optimal shapes.