Boundary geometry and asymptotics in several complex variables

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Abstract Award: DMS-0072237 Principal Investigator: David E. Barrett Professor Barrett will investigate various topics in complex analysis. One topic to be studied is the behavior of two different systems of partial differential equations implementing deformation of a real hypersurface in two-dimensional complex euclidean space by the Levi-form of the hypersurface. The systems are analogous, respectively, to harmonic-mapping heat flow and to the Ricci-flow on the space of conformal metrics, but these particular systems have special features (the role of Lorentzian geometry in the target space and the inclusion of lower-order terms which are not conjugation invariant in the source space) that introduce new phenomena and difficulties. The associated steady-state system has known applications to function theory and engineering, and the study of the time-dependent versions given above may lead to new insights into analytic continuation. A second topic to be studied is the boundary behavior of the Bergman kernel function (off the diagonal) and Bergman representative coordinates on domains with corners, with particular interest in the case of generic intersections of strictly pseudoconvex domains (the case of intersecting balls serving as a key model problem). Professor Barrett will investigate various problems involving multiple parameters (the parameters are understood to lie in the so-called complex number system, a widely-used extension of the standard number system). One topic involves the study of systems of partial differential equations which serve to flatten a surface in the parameter space; sometimes the equations push the surface to an equilibrium configuration (the situation is somewhat analogous to that of a soap film attached to a fixed wire boundary), but sometimes the surface breaks before reaching equilibrium (for example, this will happen if there is no available equilibrium configuration). The computation of equilibrium configurations for these problems (or the documentation that no equilibrium exists) is important in classical function theory, and is also a central topic in the engineering discipline known as "H-infinity control theory." A second topic to be studied is based on Stefan Bergman's method of finding a sort of "ideal form" for a region in complex multiparameter space. In the one-parameter setting Bergman's method tells us how to perform the useful task of smoothing out corners appearing in the boundary of the region (such smoothing is of fundamental importance for example in classical aerodynamics); the proposed research will examine what happens to corners in the multi-parameter setting.