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

University Of Alabama At Birmingham, Birmingham AL

Investigators

Abstract

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.