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

主要研究者的研究兴趣在于保角几何和偏微分方程组的交集。保角几何研究的是空间上的保角变换集。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是调查的中心。主要目标之一将是将微分几何中的不同子场连接起来。这一推广将极大地扩大应用范围。本课题研究的目的是利用偏微分方程来理解保角几何中的基本问题。这包括共形不变量的构造和分类问题，该问题起源于数学物理。它还包括研究共形不变量与其他几何量之间的关系，以及建立几何不等式。PI打算发展权重理论，以完善共形不变量的解析研究。她还将研究子流形上与高阶外曲率相关的全局刚性。

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.