Holder Continuous Mappings into Sub-Riemannian Manifolds

Mirra, Jacob (2018) Holder Continuous Mappings into Sub-Riemannian Manifolds. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

We develop analytic tools with applications to the study of H\"older continuous mappings into manifolds, especially sub-Riemannian manifolds like the Heisenberg Group. The first is a notion of a pullback $$f^* \kappa$$ of a differential form $$\kappa$$ by a Slobodetski\u{\i} (or fractional Sobolev) mapping $$f \in W^{s,p}(M,N)$$ between manifolds; the second is Hodge decomposition of these objects $$f^* \kappa = \Delta \omega$$; the third tool is a notion of generalized Hopf Invariant for mappings $$f : \mathbb{S}^{4n-1} \rightarrow \mathbb{H}_{2n}$$ from spheres into the Heisenberg Group, which relies on this Hodge decomposition. This latter idea was explored in \cite{hajlasz2014homotopy} for Lipschitz maps. Here, the definition is extended to H\"older continuous maps. The first tool allows an apparently simpler proof of a slight generalization of Gromov's non H\"older-embedding theorem for maps $$f \in C^{0,\gamma}(\mathbb{R}^{n+1},\mathbb{H}_n)$$, $$\gamma > \frac{n+1}{n+2}$$. The Hopf invariant allows for another rigidity result for $$\gamma$$-H\"older maps, again for sufficiently large $$\gamma$$.

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Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
Mirra, Jacobjrm152@pitt.edujrm1520000-0002-9639-6315
ETD Committee:
Committee ChairHajlasz, Piotrhajlasz@pitt.eduhajlasz
Committee CoChairSchikorra, Arminschikorra@pitt.eduschikorra
Committee MemberDeblois, Jasonjdeblois@pitt.edujdeblois
Committee MemberHales, Thomashales@pitt.eduhales
Committee MemberLeoni, Giovannigiovanni@andrew.cmu.edu0000-0002-1228-8561
Date: 27 September 2018
Date Type: Publication
Defense Date: 25 June 2018
Approval Date: 27 September 2018
Submission Date: 23 July 2018
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 127
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: sub-Riemannian geometry, Heisenberg group, H\"older mappings, Jacobian, Gromov's conjecture, Hopf invariant
Date Deposited: 27 Sep 2018 20:02