Beauchamp, Marc J.
(2017)
On Extremal Punctured Spheres.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
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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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
|
| 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 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/32702 |
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