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

本文将解决关于低维流形超曲面的平均曲率流（MCF）的几个基本开放问题，并将MCF作为解决3维流形拓扑中的核心问题的工具。特别是，PI将研究平均曲率流的正则性问题，研究超大体积3-流形的几何和拓扑结构，并利用这些结果来攻击虚拟哈肯猜想。平均曲率流和其他曲率流因其内在的美以及其自身的内在利益和它们在其他科学领域的潜在应用而发展起来，如数学金融和材料科学模型，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，表面沿表面积减小最大的方向移动，因此最小表面在MCF下保持静止。虽然已经获得了关键的基础结果，但几个最基本的问题仍未得到解答。

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.