Mathematical Sciences: Analytic and Geometric Function Theory

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Work on this project will concentrate on problems arising in the theory of several complex variables, pluripotential theory, mapping properties in several variables, characteristic classes and with the rigidity of compact minimal surfaces in spheres. Investigations into the stability of solutions and the Dirichlet problem for the complex Monge-Ampere operator will be carried out. These include determining the class of plurisubharmonic functions which make up the correct domain for this operator. Stability questions center on the dependence of solutions on both boundary data and the inhomogenous part of the operator. The extent to which solutions converge when these related elements converge is still open. Questions regarding mapping functions derive from earlier results which show that high enough infinitesimal boundary contact of holomorphic maps forces the map to be biholomorphic. Work will be done on natural extensions of this work to find n- point Pick-Nevanlinna inequalities in several variables and to establish multivariate rigidity conditions on maps whose linear part is the identity. Studies of minimal surfaces and surfaces of constant mean curvature are necessarily linked to the understanding of harmonic maps. Recent work has shown that the set of conformal structures of fixed topological type that can be realized as embedded minimal surfaces in the three-sphere is compact. Minimally immersed surfaces are finite in number. Work will be done in establishing whether or not this can be reduced to uniqueness. This research has application to nonlinear partial differential equations, differential geometry and higher dimensional potential theory.