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

这项建议将解决关于低维流形的超曲面的平均曲率流(MCF)的几个基本公开问题，并将引入MCF作为解决三维流形拓扑中的中心问题的工具。特别是，PI将研究平均曲率流的正则性问题，研究超大体积三维流形的几何和拓扑，并利用这些结果来攻击虚拟Haken猜想。平均曲率流和其他曲率流是因为它们的内在美以及它们本身的内在趣味以及它们在其他科学领域的潜在应用而发展起来的，如数学金融和材料科学，以建模例如期权定价、退火金属中的颗粒运动和晶体生长。在平均曲率流下，曲面沿着曲面面积减小最大的方向移动，因此最小曲面在MCF下保持静止。虽然已经取得了关键的基础性成果，但仍有几个最基本的问题没有得到回答。

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.