Linear and Nonlinear Laplacians on Riemannian Manifolds

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9622709 Donnelly This proposal contains three separate projects: nodal sets of eigenfunctions for higher order elliptic operators; L2 cohomology of the Bergman metric, and the Dirichlet problem at infinity for harmonic maps between rank one symmetric spaces. For the nodal set project, the investigator proposes to find a lower bound for its Hausdorff measure. For the L2 cohomology project, he intends to generalize the vanishing theorem for strictly pseudoconvex domains to include weakly pseudoconvex domains. For the Dirichlet problem, the investigator is interested in providing a counterexample to existence of harmonic maps. The main thrust of this project is the study of Laplacians on Riemannian manifolds. Given a Riemannian manifold, that is, a curved space possessing a distance function, its Laplacian determines the behavior of a very important class of functions defined on the space, namely, harmonic functions. Harmonic functions arise in many different contexts: in mathematics, they arise as pure and imaginary parts of complex-analytic functions; in applications, they describe various heat and wave phenomena.