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Geometries of Hyperbolic Surfaces with and without Boundary

Romanelli, Kimberly (2018) Geometries of Hyperbolic Surfaces with and without Boundary. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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In this dissertation, I will investigate three different points of view of maximizing packings on complete hyperbolic surfaces with finite area, possibly with geodesic boundary. This optimization takes place over the Teichmuller space of each surface.
First I find a sharp upper bound for the packing radius, and consequently the injectivity radius, of a surface with Euler characteristic chi, n cusps, and b geodesic boundary components. In particular, I do not fix these boundary lengths. This is an extension of the results found in DeBlois's papers of 2015 and 2017 to the with-boundary setting.
Second, I find a formula for maximizing the systole of loops on the three-holed sphere with fixed boundary lengths and discuss the more general claim on the general systole of loops formula asserted by Gendulphe in his pre-print "The injectivity radius of hyperbolic surfaces and some Morse functions over moduli spaces."
Finally, I present work towards proving Conjecture 1.2 in Hoffman and Purcell's paper "Geometry of planar surfaces and exceptional fillings," which asserts an upper bound of 10/sqrt{3} on the minimal area of a packing of a hyperbolic surface by horoball cusp neighborhoods, over all such packings. I verify that the geometric decorations they define cover the decorated Teichmuller space of a generic surface. I then find an explicit upper bound on the minimal area over all packings of both the three- and four-punctured spheres using these coordinates as well as the decorations which achieve this maximum.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Romanelli, Kimberlykar160@pitt.edukar160
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairDeBlois,
Committee MemberHales,
Committee MemberLennard,
Committee MemberMatthew,
Date: 26 September 2018
Date Type: Publication
Defense Date: 23 July 2018
Approval Date: 26 September 2018
Submission Date: 26 July 2018
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 88
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: topology, geometry, hyperbolic surfaces, packing
Date Deposited: 26 Sep 2018 23:08
Last Modified: 26 Sep 2018 23:08

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