Canonical metrics and geometric flows on non-compact manifolds

微分几何是研究空间、空间形状以及两者之间相互作用的学科。我的研究主要集中在两个方面。第一类涉及研究空间本身的基本结构，以及它的形状如何影响这一结构。相反，我们还研究了确定底层空间何时可以呈现出与几何对象相对应的理想形状的问题，这些几何对象具有美丽的数学描述，但对于哪些具体实例很难构造。第二类涉及研究子空间在更大空间内移动的方式。我用来研究这两类问题的主要分析工具分别是Ricci流动方程和平均曲率流动方程。里奇流动方程本质上规定了一种方法，可以将任何给定空间的形状变形为几何上更好的形状。值得注意的是，在许多情况下，这种流动实际上产生了

Differential geometry is the study of space, its shape, and the interaction between the two. My research focuses in two categories of this study. The first category involves studying the underlying fabric of space itself and how this is influenced by its shape. Conversely, we also study the problem of determining when an underlying space can assume certain ideal shapes corresponding to geometric objects with beautiful mathematical descriptions, but for which concrete examples are very hard to construct. The second category involves studying ways in which subspaces move within a larger space. The main analytic tools I use to study problems in these two categories are the Ricci flow equation and the mean curvature flow equation respectively. The Ricci flow equation essentially prescribes a way to deform the shape of any given space into a geometrically nicer one. The remarkable thing is that in many cases, the flow actually produces