Differential geometry studies the topological, geometric and analytic properties of the spaces. Since all physical events take places in a space, by Einstein's relativity the spatial metric properties of the underlying space have fundamental impacts on the physical events happening in the space, including the very one we live in. This relation places the subject of the differential geometry in a central position of the mathematics and some fundamental issues of sciences. Modern differential geometry appeals to the theory/methods of linear and nonlinear partial differential equations. The proposal involves the study of differential geometry via the method of the evolutional partial differential equations with the focus on the study of the entropy, the monotonicity, and related sharp differential estimates motives by thermodynamics and statistical mechanics. The study of Gauss curvature flow has important consequences on the subjects of the image processing, affine differential geometry and convex geometry. The study of the Ricci flow in the proposal advances the understanding of the differential topological structure of the spaces and is related to the theoretic physics with direct bearing on the high energy physics. The resolving of the problems involved in the proposal advances the above mentioned related subjects in sciences. The out-reach components of the proposal disseminate the new results to general public and contributes towards the mathematical education in the community in the greater area of San Diego and beyond. It also contributes to the training and the early career development of under-represented groups.
The technical aspects of the proposal involve the development of the sharp monotonicity of the entropy and related sharp point wise estimates. The study of differential geometric problems relies on solving evolution equations such as the Ricci flow equation, Gauss curvature flow equation or other nonlinear evolution equations, and/or the study of delicate properties of the solutions obtained. The theory of modern partial differential equations reduces the study of the solutions of various linear and nonlinear equations to the monotonicity of certain entropic quantities and related point wise estimates. The de-regularizing estimates proposed in this proposal broadens the scope of the already far-reaching gradient estimates, Hessian estimates and curvature estimates techniques developed in the last several decades to more general settings, by requiring less regularity of the solutions involved, while encoding far more geometric information. They have produced and will produce much powerful geometric consequences. The techniques developed can in turn advances the study of the related partial differential equations, which usually occupies a central role in the theory of the partial differential equations.
