Boundary geometry and asymptotics in several complex variables

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AbstractAward: DMS-0072237Principal Investigator: David E. BarrettProfessor Barrett will investigate various topics in complexanalysis. One topic to be studied is the behavior of twodifferent systems of partial differential equations implementingdeformation of a real hypersurface in two-dimensional complexeuclidean space by the Levi-form of the hypersurface. Thesystems are analogous, respectively, to harmonic-mapping heatflow and to the Ricci-flow on the space of conformal metrics, butthese particular systems have special features (the role ofLorentzian geometry in the target space and the inclusion oflower-order terms which are not conjugation invariant in thesource space) that introduce new phenomena and difficulties. Theassociated steady-state system has known applications to functiontheory and engineering, and the study of the time-dependentversions given above may lead to new insights into analyticcontinuation. A second topic to be studied is the boundarybehavior of the Bergman kernel function (off the diagonal) andBergman representative coordinates on domains with corners, withparticular interest in the case of generic intersections ofstrictly pseudoconvex domains (the case of intersecting ballsserving as a key model problem).Professor Barrett will investigate various problems involvingmultiple parameters (the parameters are understood to lie in theso-called complex number system, a widely-used extension of thestandard number system). One topic involves the study of systemsof partial differential equations which serve to flatten asurface in the parameter space; sometimes the equations push thesurface to an equilibrium configuration (the situation issomewhat analogous to that of a soap film attached to a fixedwire boundary), but sometimes the surface breaks before reachingequilibrium (for example, this will happen if there is noavailable equilibrium configuration). The computation ofequilibrium configurations for these problems (or thedocumentation that no equilibrium exists) is important inclassical function theory, and is also a central topic in theengineering discipline known as "H-infinity control theory." Asecond topic to be studied is based on Stefan Bergman's method offinding a sort of "ideal form" for a region in complexmultiparameter space. In the one-parameter setting Bergman'smethod tells us how to perform the useful task of smoothing outcorners appearing in the boundary of the region (such smoothingis of fundamental importance for example in classicalaerodynamics); the proposed research will examine what happens tocorners in the multi-parameter setting.