Mathematical Sciences: Partial Differential Equations

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Work will continue on research projects involving the interplay of mathematical analysis, geometry and algebra. Questions arising in conformal and biholomorphic geometry and Fuchsian partial differential equations will be the focus of the research. The first project is a continuing collaboration aimed at the construction of all scalar invariants of a conformal or Cauchy- Riemann manifold. A complete solution for odd dimensions in the conformal case appears to be accessible at this time. The problem has been reduced to a purely algebraic problem in invariant theory. A partial solution in even dimensions has also been found. The ultimate goal of this work is to give an explicit invariant description of the asymptotic expansion of the Bergman kernel of a smooth bounded strictly pseudoconvex domain in complex n-dimensional space. A related project involves the application of representation-theoretic methods to parabolic invariant theory. The objective is the same as above except that an alternate approach is to be analyzed using Verma modules which have the same structure as the space of jets of a conformal structure. A formulation of the problem in terms of Verma modules brings more structure and tools to bear as well as providing a large class of test problems on which to develop methods. The third project concerns the study of existence, uniqueness and regularity of Fuchsian differential equations described by Einstein metrics and harmonic maps on conformally compact manifolds. These equations are given by nonlinear systems of elliptic type which degenerate at the manifold boundary. As a first goal, work will be done in finding conformally compact metrics having constant negative Ricci curvature, with the conformal class specified in advance. This would extend the model case of the hyperbolic metric of constant negative curvature defined on a ball. Here, the conformal class is the standard metric of the sphere.//