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On Extremal Punctured Spheres

Beauchamp, Marc J. (2017) On Extremal Punctured Spheres. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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We present a classification of extremal n-punctured spheres. We show that there are exactly three such surfaces which feature multiple extremal disks: the unique extremal 3- and 4-punctured spheres and a particular 6-punctured sphere as well. We prove that each of these surfaces admit precisely two extremal disks and in all cases the disks are exchanged by a self-isometry of the surface. We demonstrate that for all other n, each extremal n-punctured sphere has a unique extremal disk. We derive formulas to count the exact number of extremal punctured spheres and determine the asymptotic growth rate of this total. Finally, we establish an upper bound on the number of once-punctured extremal surfaces by determining the precise number of extremal disk - surface pairs in this case.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Beauchamp, Marc J.mjb189@pitt.edumjb189
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairDeBlois, Jasonjdeblois@pitt.edujdeblois
Committee MemberHales, Thomashales@pitt.eduhales
Committee MemberGartside, Paulgartside@math.pitt.edugartside
Committee MemberMcReynolds,
Date: 23 September 2017
Date Type: Publication
Defense Date: 4 May 2017
Approval Date: 23 September 2017
Submission Date: 31 July 2017
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 91
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: Hyperbolic Geometry, Hyperbolic Surfaces, Extremal Surfaces, Triangulations, Injectivity Radius, Punctured Spheres
Date Deposited: 24 Sep 2017 00:01
Last Modified: 24 Sep 2017 00:01


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