Mathematical Sciences: Some Problems in Geometric Analysis

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Paul Yang will work on three central problems in geometric analysis. The first of these involves investigating which functions on the sphere are Gaussian curvature functions of metrics conformally equivalent to the standard metric. The second is to find out to what extent the universal cover of certain surfaces of negative curvature resembles the ball. In the final question the objective is to study the analytic and geometric nature of regions which give reliable data when used as sampling regions. The Nirenberg problem concerning characterizing Gaussian curvature functions is a delicate one in view of the necessary condition observed by Kazdan and Warner. Yang will focus his efforts on an investigation of functions admitting only two critical points. In the context of the uniformization problem Yang will investigate the conjecture that the universal cover of a Kahler manifold of negative sectional curvature is biholomorphically equivalent to a bounded domain in complex n-space. The Pompieu problem will be reformulated in terms of the existence of an infinite sequence of eigenfunctions to two dual systems of overdetermined boundary conditions.