Partial Differential Equations in Riemannian Geometry

黎曼几何中的偏微分方程

• 批准号: 1649174
• 负责人: Simon Brendle
• 金额: $ 25.6万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1649174&HistoricalAwards=false
• 关键词: Partial; Differential; Equations; Riemannian; Geometry

The proposed research is concerned with questions at the crossroads of geometry and partial differential equations. In recent years, methods involving partial differential equations have become increasingly important in geometry: Perhaps the most spectacular example is the Ricci flow method pioneered by Richard Hamilton and the solution of the Poincare conjecture by Grigoriy Perelman; other important advances were made in the understanding of minimal surfaces. These are surfaces which minimize area and serve as models for soap films. The goal of this project is to further our understanding of these and similar equations.Specifically, the PI and Gerhard Huisken are planning to study singularity formation for geometric flows. Besides the Ricci flow, a particular focus are flows for 2-convex hypersurfaces. The PI is also planning to work on problems in minimal surface theory. In this direction, he is planning to study the eigenvalues of the Laplace operator on a minimal surface; another direction is the classification of minimal annuli with free boundary. Finally, the PI is planning to study the gluing approach to the construction of solutions to geometric PDEs.

拟议的研究关注的问题在十字路口的几何和偏微分方程。近年来，涉及偏微分方程的方法在几何学中变得越来越重要：也许最引人注目的例子是理查德·汉密尔顿开创的里奇流方法和格里戈里·佩雷尔曼解决庞加莱猜想;其他重要的进展是在理解极小曲面方面。这些是最小化面积的表面，用作肥皂膜的模型。该项目的目标是加深我们对这些方程和类似方程的理解。具体来说，PI和Gerhard Huisken计划研究几何流动的奇异性形成。除了Ricci流之外，一个特别的焦点是2-凸超曲面的流。PI还计划研究最小曲面理论中的问题。在这个方向上，他计划研究极小曲面上的拉普拉斯算子的特征值;另一个方向是自由边界极小环的分类。最后，PI计划研究胶合方法来构建几何偏微分方程的解决方案。