Geometry of Conformal and Quasiconformal Mappings

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Abstract for DMS - 0103626The PI, Christopher Bishop, will study the geometric propertiesof conformal mappings in the plane and quasiconformal mappings in space,focusing on the expansion properties of such maps and investigatingvarious applications to geometric function theory, dynamics andtopology. The PI has shown that a result of Dennis Sullivan'sconcerning the geometry of convexbodies in hyperbolic three space implies a factorizationtheorem for conformal mappings in the plane and this, in turn,implies uniform bounds on the amount ofcontraction a conformal map in the plane can have. Finding the best constantsin the factorization theorem has consequences for wellknown problems such as dimension distortion, integral means and Brennan'sconjecture. The PI will continue his work onlimit sets of Kleinian groups, a natural and important class of fractal sets.The questions here are mainly to estimate the fractal dimension of these setsand study the behavior of the dimension as the group is deformed. The PI will also work on the metric propertiesof harmonic measures, particularly results which quantify the idea that harmonicmeasure cannot be concentrated on a small set. Problems include thelower density conjecture, stability of harmonic measure and the growth rateof diffusion limited aggregation. A few other questionsinvolving quasiconformal and biLipschitz maps are also considered.Conformal mappings are a class of functions which are importantin many area of mathematics and which are closely relatedto 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 quantifyingthis expansion by approximating conformal maps by (the moregeneral class of) quasiconformal maps and showing theseapproximations may be taken with a very strong expansion property. This has given a clearer understanding of someknown 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 computationalaspects of this connection which may lead to new methods of computing conformal maps and Greens functions (importantfor a variety of applications). The PI will also continue his investigation of other problems including the geometry of randompaths such as Brownian motion, the stability underperturbation of certain dynamical systems and fundamentalgeometric properties of conformal and quasiconformal mappings.