Quasisymmetric Maps, Doubling Measures, and Geometry of Banach Spaces

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

This project brings together topics from geometric analysis, harmonic analysis, and nonlinear functional analysis. The concept that connects the three areas is the notion of an accretive map, which can be thought of as a far-reaching generalization of an increasing function of one real variable. Some of the problems to be addressed within the project are the following: (1) bi-Lipschitz equivalence of Euclidean spaces with nonsmooth conformal metrics, in particular with metrics defined by Riesz potentials; (2) construction of quasisymmetric maps as vector-valued potentials of possibly non-doubling measures; (3) geometric properties of measures that satisfy a stronger, isotropic form of the doubling condition; (4) uniform continuity and Lipschitz continuity of approximate duality maps in Banach spaces; (5) variants of the Beurling-Ahlfors extension theorem for accretive maps and for complex gradients of real-valued functions. A fundamental problem in geometry and analysis is to find a parametrization of a given metric space (i.e., a set with a notion of distance) that makes it possible to visualize the space and to understand its structure. (One practical situation in which this issue arises is the construction of flat maps of the mammalian cerebral cortex.) The Riemann mapping theorem provides a parametrization of a surface that preserves all angles but that can significantly distort distances between points. This distortion of distances can sometimes be reduced by a carefully chosen additional deformation of the flat region onto which the surface is mapped. Our current understanding of such deformations of Euclidean spaces is far from complete even in two dimensions. The principal investigator is interested in using accretive maps to shed new light on this subject. Under an accretive map a space can be significantly stretched or shrunk at multiple locations, but it is never rotated by more than a predetermined amount.