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，在最小表面理论的许多长期存在的问题上都有突破，许多数学家的重要贡献。 最小的表面是大地学的自然概括。 它是局部最小化区域的表面。 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关于几何分析，包括几何进化方程，例如平均曲率和RICCI流以及功能理论。我们还提出了与Bruce Kleiner的联合项目，涉及平均凸集集的平均曲率流以及与Nancy Hingston的平均曲率流有关地理学的Morse Index Bounds。 最后，我们提出了一个与Camillo de Lellis有关Min-Max构造和应用的项目，以及与Camillo de Lellis和Bill Minicozzi的联合项目有关通用指标。