Nonlinear PDEs in Geometric Analysis

Missouri State University, Springfield MO

Investigators

Abstract

Abstract Award: DMS-0072328 Principal Investigator: Wenxiong Chen Professor Wenxiong Chen will work on a series of nonlinear problems in differential geometry. One of his main subjects is prescribing Gaussian and scalar curvature: given a function on a Riemannian manifold, can it be realized as the curvature of some pointwise conformal metric? This is equivalent to solving certain semi-linear elliptic partial differential equations. He will also work on uniformization theorems on singular surfaces and the existence of harmonic maps between complete, noncompact manifolds, which are closely related to the study of fourth order nonlinear elliptic equations, nonlinear elliptic systems, and nonlinear parabolic systems. According to Einstein, the Universe we live in is a curved space, in which gravity is realized as a distortion or bending of space-time in the neighborhood of a massive object and this change of shape is measured by the curvature tensor. Chen's research is focused on understanding when a given function can become a curvature. This is a challenging problem in Riemannian geometry and the analysis of partial differential equations, to which many researchers have contributed. The nonlinear partial differential equations and systems studied in this project have various applications in physics, chemistry, and biology, such as in fluid dynamics, combustion theory, and river pollution