Analysis and Geometry in Metric Spaces: Sobolev Mappings, the Heisenberg Group, and the Whitney Extension Theorem

Zimmerman, Scott (2017) Analysis and Geometry in Metric Spaces: Sobolev Mappings, the Heisenberg Group, and the Whitney Extension Theorem. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

This thesis focuses on analysis in and the geometry of the Heisenberg group as well as geometric properties of Sobolev mappings. It begins with a detailed introduction to the Heisenberg group. After, we see a new and elementary proof for the structure of geodesics in the sub-Riemannian Heisenberg group. We also prove that the Carnot-Carath\'{e}odory metric is real analytic away from the center of the group.

Next, we prove a version of the classical Whitney Extension Theorem for curves in the Heisenberg group. Given a real valued function defined on a compact set in Euclidean space, the classical Whitney Extension Theorem from 1934 gives necessary and sufficient conditions for the existence of a $C^k$ extension defined on the entire space. We prove a version of the Whitney Extension Theorem for $C^1$, horizontal curves in the Heisenberg group.

We then turn our attention to Sobolev mappings.
In particular, given a Lipschitz map from a compact subset $Z$ of Euclidean space into a Lipschitz connected metric space, we construct a Sobolev extension defined on any bounded domain containing $Z$.

Finally, we generalize a classical result of Dubovitski\u{\i} for smooth maps to the case of Sobolev mappings. In 1957, Duvovitski\u{\i} generalized Sard's classical theorem by establishing a bound on the Hausdorff dimension of the intersection of the critical set of a smooth map and almost every one of its level sets. We show that Dubovitski\u{\i}'s theorem
can be generalized to $W_{\rm loc}^{k,p}(\mathbb{R}^n,\mathbb{R}^m)$ mappings for all positive integers $k$ and $p>n$.

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Details

Item Type:	University of Pittsburgh ETD
Status:	Unpublished
Creators/Authors:	Creators Email Pitt Username ORCID Zimmerman, Scott srz5@pitt.edu srz5
ETD Committee:	Title Member Email Address Pitt Username ORCID Committee Chair Hajlasz, Piotr hajlasz@pitt.edu hajlasz Committee Member Tyson, Jeremy tyson@illinois.edu Committee Member Lennard, Christopher lennard@pitt.edu lennard Committee Member DeBlois, Jason jdeblois@pitt.edu jdeblois
Date:	2 July 2017
Date Type:	Publication
Defense Date:	3 April 2017
Approval Date:	2 July 2017
Submission Date:	30 March 2017
Access Restriction:	No restriction; Release the ETD for access worldwide immediately.
Number of Pages:	141
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:	analysis, sub-Riemannian geometry, metric space, Sobolev, extension, Heisenberg group, Whitney extension theorem
Date Deposited:	02 Jul 2017 21:07
Last Modified:	02 Jul 2017 21:07
URI:	http://d-scholarship.pitt.edu/id/eprint/31343

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• Analysis and Geometry in Metric Spaces: Sobolev Mappings, the Heisenberg Group, and the Whitney Extension Theorem. (deposited 02 Jul 2017 21:07) [Currently Displayed]

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