Mathematical Sciences: Complex Geometry and Analysis

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9408994 Burns This award continues support for mathematical research on geometric problems associated with domains and there boundaries situated in spaces of several complex variables. Two major areas of study will be undertaken. Both are concerned with interpretation of renormalized characteristic classes and their relation to analytic problems on complex manifolds with boundary. The first involves uniformization and structure of complex hyperbolic manifolds. The uniformization question is that of determining when a component of a complex manifold which is a compact, connected, spherical CR-manifold can be realized as a domain of a ball in n-complex variables modulo a properly discontinuous group of CR-automorphisms. The second line of investigation considers certain linear partial differential equations with the goal of finding analytic interpretations of characteristic numbers which are finite for Riemannian or Kahler manifolds of infinite volume. Work will also be done in an effort to prove a converse to a recent result on Grauert tubes. Namely, to show that all biholomorphisms of these manifolds are induced by isometries. Several complex variables arose at the beginning of the century as a natural outgrowth of studies of functions of one complex variable. It became clear early on that the theory differed widely from it predecessor. The underlying geometry was far more difficult to grasp and the function theory had far more affinity with partial differential operators of first order. It thus grew as a hybrid subject combining deep characteristics of differential geometry and differential equations. Many of the fundamental structures were defined in the last three decades. Current studies still concentrate on understanding these basic mathematical forms.