Fully nonlinear geometric partial differential equations and geometric flows on Riemannian manifolds

该项目的中心主题是了解黎曼流形的各种几何量之间的关系，例如体积，面积，曲率等。为了描述这些关系，人们寻求流形上的最优几何不等式。这个建议的一部分建立在PI和Guan的共同工作上，他们已经证明了一类非凸域的Aleksandrov-Fenchel quermass积分不等式。这一方向的研究将为研究微分几何中的经典等周不等式和凸几何中的Aleksandrov-Fenchel不等式提供一种创新的方法。进一步的研究将包括这些基本几何不等式在各种几何空间。本课题第二部分研究了一般黎曼流形的曲率测量规定性问题，是PI与关鹏飞、李燕燕在欧几里德空间曲率测量规定性问题上的推广。这些研究课题是对经典微分几何中的Christofel-Minkowski问题和Aleksandrov问题的推广。本文的第三部分是基于Ricci流的熵泛函和微分Harnack不等式的研究。Ricci孤子方程、各种熵泛函沿Ricci流的单调性公式及其与局部微分哈纳克不等式的关系是高维黎曼流形中Ricci流研究的重要组成部分。本课题旨在研究微分几何中由子流形和一般黎曼流形上的完全非线性椭圆型和抛物型方程引起的问题。例如，超曲面流动方程、规定曲率测量方程、里奇流动方程等。感兴趣的问题是在微分几何和偏微分方程理论的十字路口。目的是更好地理解这些重要几何偏微分方程的几何量和性质之间的关系。

The central theme of this project is to understand the relation between various geometric quantities of Riemannian manifolds, e.g. volume, area, curvature, etc. To describe these relations, one seeks for optimal geometric inequalities on manifolds. Part of this proposal builds on the joint work of the PI and Guan who have proven the Aleksandrov-Fenchel quermassintegral inequalities for a class of non-convex domains. The research along this direction will provide an innovative method to study the classical isoperimetric inequalities from differential geometry and the Aleksandrov-Fenchel inequalities from convex geometry. Further study will include these fundamental geometric inequalities in various geometric spaces. The second part of this project studies the prescribing curvature measure problems in general Riemannian manifolds which is an extension of the joint work of the PI with Pengfei Guan and Yanyan Li on prescribing curvature measure problem in Euclidean space. These research topics are generalizations to the Christofel-Minkowski problem, and Aleksandrov problem from classical differential geometry. The third part of this proposal is based on the study of entropy functionals and differential Harnack inequality for Ricci flow. Ricci soliton equations, monotonicity formulas of various entropy functionals along Ricci flow, and their relations with local differential Harnack inequality consist important parts of the study of Ricci flow in higher dimension Riemannian manifolds. This project aims at studying problems arising from differential geometry via fully nonlinear elliptic and parabolic equations on submanifolds and general Riemannian manifolds. For example, hypersurface flow equations, prescribing curvature measure equations, Ricci flow equations, and etc. The interested questions are at the cross roads of differential geometry and the theory of partial differential equations. The goal is to better understand relations between geometric quantities and properties of these important geometric PDEs.