The first part of the proposed research concerns what happens to a hypersurface under the mean curvature flow. We are particularly interested in hypersurfaces that are in general or generic position before the flow start. The mean curvature flow is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time. Before it becomes extinct, topological changes can occur as it goes through singularities. Thus, in some sense, the topology is encoded in the singularities. The proposed project further develops the theory of generic mean curvature flow that the PI has initiated with Minicozzi. We have already classified all generic singularities and one of the main task now is to completely understand the flow itself and show that indeed a generic hypersurface only go through generic singularities. The first key point is to prove a stable manifold theorem, that is, to prove that in a neighborhood of an unstable self-shrinker the stable manifold is contained in a hyper-graph. We expect that these results will have a number of applications and plan to pursue them. A second part concerns manifolds and spaces with Ricci curvature bounds and give some new estimates for these manifolds and various applications of these estimates, in particular to Einstein metrics. A third part concerns representations of isometry groups and fundamental groups of open manifolds into general linear groups of finite dimensional vector spaces. The final and smaller part is about bounds for nodal sets (or zero sets) of eigenfunctions.
When a surface evolves over time by locally moving it in the direction where the area decreases the fastest it is said to be moving by mean curvature flow. Mathematically, this leads to a nonlinear partial differential equation which is formally similar to the equation that governs the flow of heat in physics. Moving interfaces occur in a wide range of scientific and engineering applications. Mean curvature flow and other geometric flows were developed for their intrinsic beauty as well as their potential applications to other fields to model, for instance, option pricing, motion of grains in annealing metals, and crystal growth. While key foundational results have been obtained, several of the most basic questions remain unanswered. Many applications are expected both within and outside mathematics.
