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

FRG：协作研究：平均曲率流作为低维拓扑的工具

• 批准号: 0854969
• 负责人: David Gabai
• 金额: $ 43.49万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0854969&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Mean; curvature

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.

该提案将解决有关低维流形超曲面平均曲率流 (MCF) 的几个基本开放问题，并将引入 MCF 作为解决 3 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规律性问题，研究超大体积3流形的几何和拓扑，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的开发是因为它们的内在美、它们本身的内在兴趣以及它们在其他科学领域的潜在应用，例如数学金融和材料科学，以建模，例如期权定价、谷物运动用于金属退火和晶体生长。 在平均曲率流下，表面沿表面积减小最多的方向移动，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性结果，但一些最基本的问题仍未得到解答。