Quasiconformal homeomorphisms on Cauchy-Riemann manifolds

Puqi Tang, Purdue University

Abstract

A smooth 3-manifold M is called a contact manifold if there is a nowhere integrable hyperplane distribution HM in the tangent space. A Cauchy-Riemann (CR) 3-manifold is a contact 3-manifold endowed with a complex structure on HM. It is one of the main objects studied in analysis of several complex variables. A homeomorphism between two CR 3-manifolds is called quasiconformal if, roughly speaking, it preserves the underlying contact structures and distorts the CR structures in a bounded way. In a class of homeomorphisms between two CR 3-manifolds a quasiconformal homeomorphism with minimal conformal distortion is said to be extremal. Hence an extremal quasiconformal homeomorphism among all homeomorphisms between two CR manifolds describes the nonisomorphism of these CR manifolds. The main objects in this thesis are quasiconformal homeomorphisms between strongly pseudoconvex CR 3-manifolds, especially the extremality of such homeomorphisms. This thesis proposes an analytic definition of quasiconformal homeomorphisms on CR 3-manifolds and proves all conformal homeomorphisms between embeddable CR manifolds with $L\sbsp{\rm loc}{1}$ horizontal derivatives are actually CR and smooth. Then we introduce the moduli of Legendrian curve families to estimate the conformal distortion of a homeomorphism. We give a partial solution in the aspect of regularity to the question if moduli of curve families can be used to define quasiconformal homeomorphisms. We construct extremal mappings in some homotopy classes of homeomorphisms between CR circle bundles over flat tori. These extremal mappings have behavior analogous to Teichmuller mappings on Riemann surfaces. The construction depends on some sub-Riemannian geometry and an ergodic length-volume argument developed by Teichmuller and Ahlfors on Riemann surfaces. First and second variation of conformal distortion are developed in order to test extremality of a quasiconformal homeomorphism. On CR manifolds with a free transversal circle actions, we characterize equivariant quasiconformal homeomorphisms. On $S\sp3$, which is a canonical model of such symmetric CR manifolds, we construct a large family of invariant CR structures so that the extremal mappings among the equivariant mappings between them and the standard structure are completely determined. These CR structures also serve as examples showing that the extremal quasiconformal homeomorphisms between two invariant CR manifolds are not necessarily equivariant. Above all, this thesis initiates a Teichmuller type theory on CR 3-manifolds to study the moduli problem of such manifolds. The classical Teichmuller theory gives a solution to the moduli problem of Riemann surfaces. We expect this approach will lead to complete understanding of CR 3-manifolds.

Degree

Ph.D.

Advisors

Lempert, Purdue University.

Subject Area

Mathematics

COinS