FRG: Collaborative Research: Mean curvature flow as a tool in low dimensional topology

这个建议将解决几个基本的开放问题平均曲率流（MCF）的超曲面的低维流形，并将介绍MCF作为一种工具，以解决中心问题，在3-流形拓扑。 特别是，PI将研究平均曲率流的正则性问题，研究超大体积3-流形的几何和拓扑，并使用这些结果来攻击虚拟哈肯猜想。 平均曲率流以及其他曲率流已经发展为它们的内在美以及它们自己的内在利益和它们在其他科学领域的潜在应用，如数学金融和材料科学建模，例如，期权定价，退火金属中的晶粒运动和晶体生长。 在平均曲率流下，曲面沿表面积减小最多的方向移动，因此极小曲面在MCF下保持静止。虽然已经取得了关键的基础性成果，但一些最基本的问题仍然没有答案。

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.