Nonlinear PDEs in Geometric Analysis

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AbstractAward: DMS-0072328Principal Investigator: Wenxiong ChenProfessor Wenxiong Chen will work on a series of nonlinearproblems in differential geometry. One of his main subjects isprescribing Gaussian and scalar curvature: given a function on aRiemannian manifold, can it be realized as the curvature of somepointwise conformal metric? This is equivalent to solving certainsemi-linear elliptic partial differential equations. He willalso work on uniformization theorems on singular surfaces and theexistence of harmonic maps between complete, noncompactmanifolds, which are closely related to the study of fourth ordernonlinear elliptic equations, nonlinear elliptic systems, andnonlinear parabolic systems.According to Einstein, the Universe we live in is a curved space,in which gravity is realized as a distortion or bending ofspace-time in the neighborhood of a massive object and thischange of shape is measured by the curvature tensor. Chen'sresearch is focused on understanding when a given function canbecome a curvature. This is a challenging problem in Riemanniangeometry and the analysis of partial differential equations, towhich many researchers have contributed. The nonlinear partialdifferential equations and systems studied in this project havevarious applications in physics, chemistry, and biology, such asin fluid dynamics, combustion theory, and river pollution