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 以这种方式塑造一个理想的造型！另一方面，平均曲率流规定了一种规范的方式来变形较大空间中的子空间，在这里，人们也经常以这种方式到达理想的位于较大空间中的空间。这些方程属于一类被称为几何演化方程的方程，以这种方式(以稍微更一般的方式)研究微分几何的方法被称为几何分析。从技术上讲，这些方程是分析中经典研究的偏微分方程式。尤其是由于G.Perelman对Ricci流的开创性工作及其在证明Poincar猜想中的应用，Ricci流最近受到了广泛的关注。尽管人们对这些几何演化方程越来越感兴趣，但在这个令人兴奋和肥沃的领域，加拿大的研究人员仍然相对较少。我的研究是几何分析在数学界和加拿大发展的关键组成部分。

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 an ideal shape in this way! On the other hand, the mean curvature flow prescribes a canonical way in which to deform a subspace within a larger space, and here too one often arrives at a space sitting ideally within the larger space in this way. The equations belongs to a class of equations known as geometric evolution equations, and the study of Differential geometry in this way (and in a slightly more general ways) is known as geometric analysis. These equations are technically a partial differential equation which are classically studied in analysis. The Ricci flow in particular has recently received widespread attention especially due to ground breaking work of G. Perelman on the Ricci flow and its use in proving the Poincar\'{e} conjecture. Despite the growing interest in these geometric evolutions equations, there are still relatively few Canadian researchers in this exciting and fertile area. My research is a key component of the development of geometric analysis in both the mathematical community and in Canada.