Geometry of Conformal and Quasiconformal Mappings

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

Abstract for DMS - 0103626 The PI, Christopher Bishop, will study the geometric properties of conformal mappings in the plane and quasiconformal mappings in space, focusing on the expansion properties of such maps and investigating various applications to geometric function theory, dynamics and topology. The PI has shown that a result of Dennis Sullivan's concerning the geometry of convex bodies in hyperbolic three space implies a factorization theorem for conformal mappings in the plane and this, in turn, implies uniform bounds on the amount of contraction a conformal map in the plane can have. Finding the best constants in the factorization theorem has consequences for well known problems such as dimension distortion, integral means and Brennan's conjecture. The PI will continue his work on limit sets of Kleinian groups, a natural and important class of fractal sets. The questions here are mainly to estimate the fractal dimension of these sets and study the behavior of the dimension as the group is deformed. The PI will also work on the metric properties of harmonic measures, particularly results which quantify the idea that harmonicmeasure cannot be concentrated on a small set. Problems include the lower density conjecture, stability of harmonic measure and the growth rate of diffusion limited aggregation. A few other questions involving quasiconformal and biLipschitz maps are also considered. Conformal mappings are a class of functions which are important in many area of mathematics and which are closely related to mnay physical problems (fluid flow, heat conduction, electric fields, random growth models, ...) and have been intensively studied for many years. One of the fundamental properties of such maps is expansion; they tend to push points farther apart on average. Making this precise has motivated much research in mathematical analysis. The PI has discovered a new way of quantifying this expansion by approximating conformal maps by (the more general class of) quasiconformal maps and showing these approximations may be taken with a very strong expansion property. This has given a clearer understanding of some known results and has led to progress on new problems. In particular, it implies new results about Kleinian groups (these are important examples of conformal dynamical systems, and hence a contribution to the more general area of dynamical systems, fractals and chaos). The PI's approach also ties the behavior of conformal maps to the geometry three dimensional hyperbolic space; this connection seems to be new and should lead to many interesting problems and more interaction between the areas of complex analysis and three dimensional topology (already connected in other ways). He will also investigate the computational aspects of this connection which may lead to new methods of computing conformal maps and Greens functions (important for a variety of applications). The PI will also continue his investigation of other problems including the geometry of random paths such as Brownian motion, the stability under perturbation of certain dynamical systems and fundamental geometric properties of conformal and quasiconformal mappings.