Conformal Densities and Deformations of Uniform Loewner Metric Spaces
For a conformal density defined on a bounded uniform metric measure space with bounded geometry and uniformly perfect boundary, we find a list of conditions equivalent to the condition that the deformed space is uniform. We establish that a locally quasiconvex doubling abstract domain is uniform if it has the min-max property. Given a quasisymmetric map from a locally quasiconvex Ahlfors regular space onto an Ahlfors regular space we define a density function as the ratio of the distance of the image of a point to the boundary and the distance of the pre-image to the boundary. We show that such a density is conformal. We prove that the image of the space under the quasisymmetric map is quasisymmetrically equivalent to the deformed space. We examine how the induced metric relates to the pull-back of the inner diameter distance and study the relationship between the quasihyperbolization of the deformed space and the quasihyperbolization of the image of the space under the map.
School:University of Cincinnati
School Location:USA - Ohio
Source Type:Master's Thesis
Keywords:conformal density uniform spaces loewner quasisymmetry quasiconofrmal
Date of Publication:01/01/2008