Problems in Geometric Analysis: Harmonic Maps and Holomorphic Vector Bundles

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The first project deals with energy minimizing maps to the Weil-Petersson completion of Teichmueller space. The proposed research will establish sufficient regularity for higher dimensional domains so that harmonic map theory may be used to answer rigidity questions. The second part of the proposal continues work on the Yang-Mills flow on higher dimensional Kaehler manifolds. A special focus is a comparison of the analytic singularities which occur along the flow with the algebraic singularities associated to Harder-Narasimhan filtrations of holomorphic vector bundles. The third project consists of topics related to representation varieties of surface groups. Included is a study of properties of the energy functional on Teichmueller space defined by harmonic maps associated to surface group representations. Results in this direction will have implications for the dynamics of the mapping class group action on the moduli space of representations.A significant branch of mathematical inquiry has been the relationship between the geometric, analytic, and algebraic properties of manifolds. Manifolds are higher dimensional generalizations of curves and surfaces, and they appear in a variety of situations in pure and applied mathematics. The research projects in this proposal will further our understanding of some of these objects. The equations studied -- energy minimizing maps and the Yang-Mills flow -- have their origins in the mathematical description of the physical world. They are therefore of great importance to both mathematicians and physicists.