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: 0854767
• 负责人: David Gabai
• 金额: $ 6.3万
• 依托单位: American Institute of Mathematics
• 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 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0854767&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的高空曲面平均曲率流（MCF）的几个基本开放问题，并将将MCF作为解决3个Manifold拓扑中心问题的工具。 特别是，PI将研究平均曲率流的规律性问题，研究超大卷3型构架的几何形状和拓扑结构，并使用这些结果来攻击虚拟的Haken猜想。 为了固有的美感以及其自身的固有兴趣以及对其他科学领域的潜在应用，例如数学金融和材料科学在模型上，例如期权定价，退火金属中的运动和晶体生长。 在平均曲率流下，表面沿表面积减少最大的方向移动，因此在MCF下保持最小的表面保持静态。尽管已经获得了关键的基础结果，但一些最基本的问题仍未得到解决。