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Title:Lipschitz and Holder mappings into jet space Carnot groups
Author(s):Jung, Derek
Director of Research:Tyson, Jeremy
Doctoral Committee Chair(s):Wu, Jang-Mei
Doctoral Committee Member(s):Fernandes, Rui; Kaufman, Robert
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Sub-Riemannnian Geometry
Jet spaces
Carnot groups
Geometric Analysis
Abstract:For $k,n\ge 1$, the jet space $J^k(\R^n)$ is the set of $k^{th}$-order Taylor polynomials of functions in $C^k(\R^n)$. Warhurst constructs a Carnot group structure on $J^k(\R^n)$ such that the jets of functions in $C^{k+1}(\R^n)$ are horizontal. Like in all Carnot groups, one can define a Carnot-Carath\'eodory metric on $J^k(\R^n)$ by minimizing lengths of horizontal paths. Unfortunately, exact forms or even the regularities of geodesics connecting generic pairs of points are not known for $J^k(\R^n)$. After describing the Carnot group structure of $J^k(\R^n)$, we will prove that there exists a biLipschitz embedding of $\mathbb{S}^n$ into $J^k(\R^n)$ that does not admit a Lipschitz extension to $\mathbb{B}^{n+1}$. This strengthens a result of Rigot and Wenger \cite{RW:LNE} and generalizes a result for $\mathbb{H}^n$ of Dejarnette, Haj{\l}asz, Lukyanenko, and Tyson. We will then consider a problem related to Gromov's conjecture on the H\"older equivalence of Carnot groups. We will prove that for all $m\ge 2$ and $\epsilon>0$, there does not exist an injective, locally $(\frac{1}{2}+\epsilon)$-H\"older mapping $f:\R^m\to J^k(\R)$ that is locally Lipschitz as a mapping into $\R^{k+2}$. This builds on a result of Balogh, Haj{\l}asz, and Wildrick for $\mathbb{H}^n$. We will conclude by proposing analogues of horizontal and vertical projections for $J^k(\R)$. We prove Marstrand-type results for these mappings. This continues efforts of Balogh, Durand-Cartagena, F\"assler, Mattila, and Tyson over the past decade to prove Marstrand-type theorems in a sub-Riemannian setting. We will study the metric structure of $J^k(\R^n)$, focusing primarily on the model filiform jet spaces $J^k(\R)$.
Issue Date:2018-11-08
Rights Information:Copyright 2018 Derek Jung
Date Available in IDEALS:2019-02-06
Date Deposited:2018-12

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