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.

首席研究员的研究兴趣在于共形几何和偏微分方程的交叉。保角几何是研究空间上保角变换的集合。其中一个重点将放在所谓的共形不变量上，从物理学家的角度来看，它构成了一个重要的机制，与广义相对论的基本原理有着深刻的联系。在生物和物理科学中广泛出现的质量集中现象也将成为调查的中心。主要目标之一是将微分几何中的不同子领域联系起来。这种概括将大大扩大应用的范围。本课题旨在利用偏微分方程来理解保形几何的基本问题。这包括构造和分类共形不变量的问题，它起源于数学物理。它还包括研究保形不变量与其他几何量之间的关系，建立几何不等式。PI打算发展权理论，以完善共形不变量的解析研究。她还将研究与子流形上高阶外在曲率相关的全局刚度。