Higher Order Elliptic Operators and Applications to Problems in Conformal Geometry

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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.