Research in Geometric Analysis

Washington University, Saint Louis MO

Investigators

Abstract

ABSTRACT: Krantz proposes to study the groups of holomorphic self-mappings (automorphism groups) of domains in complex space using techniques of real and complex analysis, differential geometry, Lie theory, and partial differential equations. The goal is to identify invariants that will help to classify domains up to biholomorphic equivalence. In another direction, Krantz will study real-variable Hardy spaces on domains, pursuant to earlier collaborative work with Chang and Stein. Finally, Krantz will collaborate with Marco Peloso in his continuing study of the inhomogeneous Cauchy-Riemann equations in the Sobolev topology. Applications to mapping theory and the study of the worm domain are being developed. Professor Krantz will study aspects of complex analysis related to symmetry. The notion of symmetry is measured by means of mappings of domains. The set of such mappings of a region in space to itself has an algebraic structure which is called a ``group''. The properties of this group reveal geometric structure of the region under study and vice versa. This idea of attaching an algebraic invariant to a geometric object is one of the big thrusts in twentieth-century mathematics, and uncovers new depth and texture in the subject. Krantz's study of these ``automorphism groups'' complements his continuing study of certain partial differential equations on these spatial regions. The partial differential equations can be used to construct mappings, and to calculate their regularity properties. As an ancillary to these two projects, Krantz will study certain spaces of functions and mappings which are called Hardy spaces, and whose existence was inspired by the mapping questions. The geometric study of complex function theory in several variables has applications in cosmology (thanks to work of Penrose) and in geophysics.