Geometric Analysis; Minimal Surfaces, Geometric Flows, and Function Theory

几何分析；

• 批准号: 0606629
• 负责人: Tobias Colding
• 金额: $ 59.91万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0606629&HistoricalAwards=false
• 关键词: Geometric; Analysis; Minimal; Surfaces; Flows

Recent years have seen breakthroughs on many long-standing problems in the theory of minimal surfaces, with important contributions from many mathematicians. A minimal surface is a natural generalization of geodesics. It is a surface that locally minimize area. The lamination theorem, the one-sided curvature estimate, the resolution of the Calabi-Yau conjectures for embedded surfaces, and the structure results for fixed genus surfaces (all joint work of the PI and Bill Minicozzi) have played a key role in recent breakthroughs and have been used by many people.The PI proposes, jointly with Bill Minicozzi, to continue our investigations on minimal surfaces and related areas of geometric analysis, including geometric evolution equations such as the mean curvature and Ricci flow, and on function theory.We also propose a joint project with Bruce Kleiner about mean curvature flow of mean convex sets and a joint project with Nancy Hingston about Morse index bounds of geodesics. Finally, we propose a project with Camillo De Lellis about Min-Max constructions and applications and a joint project with Camillo De Lellis and Bill Minicozzi on generic metrics.

近年来，极小曲面理论中的许多长期存在的问题都得到了突破，许多数学家做出了重要贡献。 极小曲面是测地线的自然推广。 它是局部最小化面积的曲面。 嵌入曲面的分层定理、单侧曲率估计、Calabi-Yau图的分解以及固定亏格曲面的结构结果（PI和Bill Minicozzi的所有联合工作）在最近的突破中发挥了关键作用，并已被许多人使用。PI建议与Bill Minicozzi一起继续我们对极小曲面和几何分析相关领域的研究，包括几何演化方程如平均曲率和Ricci流，以及函数论，我们还提出了与布鲁斯·克莱纳关于平均凸集的平均曲率流和与南希·欣斯顿关于测地线的莫尔斯指数界的联合方案。 最后，我们提出了一个项目与Camillo De Lellis关于最小最大的建设和应用程序和一个联合项目与Camillo De Lellis和比尔Minicozzi的通用指标。