Canonical metrics and geometric flows on non-compact manifolds

点击翻译按钮获取中文摘要

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.