Metric geometry of the Grushin plane and generalizations

Metric space;Bi-Lipschitz embedding;Sub-Riemannian geometry;Quasiconformal mapping

Romney, Matthew

关键词: Metric space;    Bi-Lipschitz embedding;    Sub-Riemannian geometry;    Quasiconformal mapping;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/99345/ROMNEY-DISSERTATION-2017.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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.