## Doctoral Dissertations

Dissertations that have an embargo placed on them will not be available to anyone until the embargo expires.

#### Author ORCID Identifier

0000-0002-4509-7091

#### AccessType

Open Access Dissertation

dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

2022

September

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