This proposal will address several fundamental open questions about mean curvature flow (MCF) of hypersurfaces of low dimensional manifolds and will introduce the MCF as a tool to address central questions in 3-manifold topology. In particular, the PI's will study regularity problems for the mean curvature flow, investigate the geometry and topology of ultra large volume 3-manifolds and use these results to attack the virtual Haken conjecture.
Mean curvature flow as well as other curvature flows have been developed for their intrinsic beauty as well as their own intrinsic interest and their potential applications to other scientific fields, like mathematical finance and material science to model, for instance, option pricing, motion of grains in annealing metals, and crystal growths. Under the mean curvature flow, surfaces move in the direction where the surface area decreases the most, thus minimal surfaces remain static under the MCF. While key foundational results have been obtained, several of the most basic questions remain unanswered.
