Higher Order Elliptic Operators and Applications to Problems in Conformal Geometry

Princeton University, Princeton NJ

Investigators

Abstract

Chang proposes to continue her study of problems in conformal geometry involving higher order elliptic operators and systems. The main theme is to study the integrand of the Gauss-Bonnet formula on 4-manifolds via a fourth order elliptic operator with leading term the bi-Laplace operator called the Paneitz operator. Three research projects are proposed. The first one connects the study of the integrand to the study of Monge-Ampere equations. The second one is to extend her earlier work on the regularity of bi-harmonic maps to a general setting. The third one is to generalize the classical work of Cohn-Vossen and Huber relating the growth of the Gaussian curvature to the Euler characteristic from complete surfaces to complete 4-manifolds. The main motivation of the project is to study the geometric and topological structure of manifolds via analytic methods--mainly via the study of the partial differential equations satisfied by the curvature of the manifold.