On Thurston's parametrization of \(\mathbb{C}{\rm P}^1\)-structures

Thurston related \(\mathbb{C}{\rm P}^1\)-structures (complex projective structures) and equivariant pleated surfaces in the hyperbolic-three space \(\mathbb{H}^3\), in order to give a parameterization of the deformation space of \(\mathbb{C}{\rm P}^1\)-structures. In this note, we summarize Thurston...

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Bibliographic Details
Published inarXiv.org
Main Author Baba, Shinpei
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 08.06.2020
Subjects
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Parameterization
Theorems
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Summary:Thurston related \(\mathbb{C}{\rm P}^1\)-structures (complex projective structures) and equivariant pleated surfaces in the hyperbolic-three space \(\mathbb{H}^3\), in order to give a parameterization of the deformation space of \(\mathbb{C}{\rm P}^1\)-structures. In this note, we summarize Thurston's parametrization of \(\mathbb{C}{\rm P}^1\)-structures, based on Kamishima-Tan and Kulkani-Pinkall. We, in addition, give independent proofs for the following well-known theorems on \(\mathbb{C}{\rm P}^1\)-structures by means of pleated surfaces given by the parameterization. (1) Goldman's Theorem on \(\mathbb{C}{\rm P}^1\)-structures with quasi-Fuchsian holonomy. (2) The path lifting property of developing maps in their domains of discontinuity in \(\mathbb{C}{\rm P}^1\).
ISSN:2331-8422