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Author ORCID Identifier
https://orcid.org/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
Recommended Citation
Nakamura, Tetsuya, "A Representation for CMC 1 Surfaces in H^3 Using Two Pairs of Spinors" (2022). Doctoral Dissertations. 2698.
https://doi.org/10.7275/31050475
https://scholarworks.umass.edu/dissertations_2/2698