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Author ORCID Identifier

0000-0002-4509-7091

AccessType

Open Access Dissertation

Document Type

dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

Year Degree Awarded

2022

Month Degree Awarded

September

First Advisor

Franz Pedit

Subject Categories

Geometry and Topology

Abstract

For Bryant's representation $\Phi\colon \widetilde{M} \rightarrow \SL_2(\C)$ of a constant mean curvature (CMC) $1$ surface $f\colon M\rightarrow \Hyp^3$ in the $3$-dimensional hyperbolic space $\Hyp^3$, we will give a formula expressed only by the global $\tbinom{P}{Q}$ and local $\tbinom{p}{q}$ spinors and their derivatives. We will see that this formula is derived from the Klein correspondence, understanding $\Phi$ as a null curve immersion into a $3$-dimensional quadric. We will show that, if $f$ is a CMC $1$ surface with smooth ends modeled on a compact Riemann surface, the linear change of $\tbinom{P}{Q}\oplus \tbinom{p}{-q}$ by some $\Sp(\C^4)$ matrices gives rise to a transformtion of the surface to new CMC $1$ surfaces with smooth ends, which have the same Willmore energy. As an example, we will consider a periodic deformation of catenoid cousins with smooth ends acted by a real one-parameter subgroup of $\Sp(\C^4)$. We will apply our construction methods using the global and local spinors to investigate the space of genus $0$ CMC $1$ surfaces with smooth ends with prescribed (small) number of ends and (low) Willmore energy. We will explicitly describe the space of all genus $0$ CMC $1$ surfaces with smooth ends of Willmore energy $16\pi$. We will also reduce the existence problem of genus $0$ CMC $1$ surfaces with three smooth ends to the existence of solutions of an algebraic equation system.

DOI

https://doi.org/10.7275/31050475

Available for download on Friday, September 01, 2023

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