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Title:Metric geometry of the Grushin plane and generalizations
Author(s):Romney, Matthew
Director of Research:Tyson, Jeremy T.
Doctoral Committee Chair(s):Wu, Jang-Mei G
Doctoral Committee Member(s):D'Angelo, John P.; Leininger, Christopher J.
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Metric space
Bi-Lipschitz embedding
Sub-Riemannian geometry
Quasiconformal mapping
Abstract:Given $\alpha>0$, the $\alpha$-Grushin plane is $\mathbb{R}^2$ equipped with the sub-Riemannian metric generated by the vector fields $X = \partial_1$ and $Y = |x_1|^{\alpha} \partial_2$. It is a standard example in sub-Riemannian geometry, as a space which is Riemannian except on a small singular set---here the vertical axis, where the vector field $Y$ vanishes. The main purpose of this thesis is to study various problems related to the metric geometry of the $\alpha$-Grushin plane and a generalization of it, termed {\it conformal Grushin spaces}. One such problem is the embeddablity of these spaces in some Euclidean space under a bi-Lipschitz or quasisymmetric mapping. Building on work of Seo \cite{Seo:11} and Wu \cite{Wu:15}, we prove a sharp embedding theorem for the $\alpha$-Grushin plane and a general embedding theorem for conformal Grushin spaces under appropriate hypotheses. We also study quasiconformal homeomorphisms of the $\alpha$-Grushin plane. In the final section, we solve a separate problem regarding quasiconformal mappings in metric spaces. The main result states that if a metric space homeomorphic to $\mathbb{R}^2$ can be quasiconformally parametrized by a domain in $\mathbb{R}^2$, then one can find a mapping which improves the dilatation to within a universal constant. A non-sharp theorem of this type was recently proved by Rajala; our theorem gives the sharp bounds for this problem.
Issue Date:2017-12-03
Rights Information:Copyright 2017 Matthew Romney
Date Available in IDEALS:2018-03-13
Date Deposited:2017-12

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