Fully Nonlinear Geometric Partial Differential Equations

University Of California-Irvine, Irvine CA

Investigators

Abstract

This project concerns investigations of fundamental problems at the interface of differential geometry, partial differential equations and theoretical physics. The fundamental laws of nature are described in the language of mathematics using ideas from differential geometry and partial differential equations. Understanding the behavior of solutions to differential equations is an important part of understanding the structure of the universe. This proposal concerns a centuries-old class of mathematical objects called Monge-Ampere equations. These equations arise naturally in the study of geometry and are closely related to Einstein's equation in general relativity and the Hull-Strominger system from string theory. A main goal of this project is to develop analytical techniques to investigate the important properties of the solutions to this type of equations, which will lead to deep understanding of the fundamental geometric structures. The study requires a broad range of tools from real and complex analysis, as well as algebraic and differential geometry. Progress on these questions will not only shed some light on some basic problems in mathematics, but will also have applications in physics and other sciences. This project will investigate the interaction between fully nonlinear partial differential equations and complex geometry. In particular, the investigator will study the solvability of complex Monge-Ampere type equations deduced from the study of a generalized Hull-Strominger system in string theory, which can be viewed as a generalization of Ricci-flat metrics on non-Kahler complex manifolds. Building on his joint work with Phong and Picard, the investigator will develop new analytic techniques for further understanding of this type of equation and to provide a complete answer to the question raised by Fu and Yau. Another major goal of this project is to study the anomaly flow, which was introduced by the investigator and his coauthors, aiming to develop analytical methods for solving the Hull-Strominger system on general three dimensional non-Kahler Calabi-Yau manifolds. Short-time existence of solutions of the flow were obtained in the joint work with Phong and Picard. The investigator will study the long-time behavior and convergence of this flow, and will also use this flow investigate the relation between the balanced cone and the Kahler cone on Kahler manifolds. To accomplish these goals, the investigator will develop new tools for the study of nonlinear elliptic and parabolic partial differential equations without concavity property. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.