Geometric Analysis in Conformal Geometry and Fully Nonlinear Elliptic Partial Differential Equations

共形几何和全非线性椭圆偏微分方程中的几何分析

• 批准号: 1612015
• 负责人: Yi Wang
• 金额: $ 18.05万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1612015&HistoricalAwards=false
• 关键词: Geometric; Analysis; Conformal; Geometry; Fully

The principal investigator's research interest lies at the intersection of conformal geometry and partial differential equations. Conformal geometry is the study of the set of angle-preserving transformations on a space. One focus will be on so-called conformal invariants, which form an important machinery from physicists' point of view and have deep connection to fundamental principles in general relativity. The mass concentration, a phenomenon that has widely appeared in biological and physical sciences, will also be at the center of the investigation. One of the main goals will be to connect different subfields in differential geometry. The generalization will significantly enlarge the scope of the applications.The project research aims to understand basic questions in conformal geometry by using partial differential equations. This includes problems of constructing and classifying conformal invariants, which originated from mathematical physics. It also includes studying the relationship between conformal invariants and other geometric quantities, and establishing geometric inequalities. The PI intends to develop weight theory in order to refine the analytic study of conformal invariants. She will also investigate global rigidity associated to extrinsic curvatures of higher orders on submanifolds.

主要研究者的研究兴趣在于保形几何和部分微分方程的交集。共形几何形状是对空间上一组垂直角度转换的研究。一个重点将放在所谓的保形不变板上，从物理学家的角度来看，这形成了重要的机制，并与一般相对论中的基本原理有着深厚的联系。质量浓度是一种在生物学和物理科学中广泛出现的现象，也将是研究的中心。主要目标之一是连接差异几何形状的不同子场。该项目研究的旨在通过使用部分微分方程来理解共形几何形状中的基本问题。这包括构建和分类保形不变的问题，该问题源自数学物理学。它还包括研究保形不变剂与其他几何量之间的关系，并确定几何不平等。 PI打算开发体重理论，以完善对形式不变的分析研究。她还将调查与较高订单的外部曲线相关的全球刚度。