 On Thurston’s Parameterization of ℂ P 1 $$\mathbb {C}\mathrm {P}^1$$ -StructuresShinpei Baba ()
Chapter Chapter 6 in In the Tradition of Thurston, 2020, pp 241-254 from Springer Abstract: Abstract Thurston established a correspondence between ℂ P 1 $$\mathbb {C}\mathrm {P}^1$$ -structures (complex projective structures) and equivariant pleated surfaces in the hyperbolic-three space ℍ 3 $$\mathbb {H}^3$$ , in order to give a parameterization of the deformation space of ℂ P 1 $$\mathbb {C}\mathrm {P}^1$$ -structures. In this note, we summarize Thurston’s parametrization of ℂ P 1 $$\mathbb {C}\mathrm {P}^1$$ -structures, based on [15] and [17], giving an outline and the key points of its construction. In addition we give independent proofs for the following well-known theorems on ℂ P 1 $$\mathbb {C}\mathrm {P}^1$$ -structures by means of pleated surfaces given by the parameterization. (1) Goldman’s Theorem on ℂ P 1 $$\mathbb {C}\mathrm {P}^1$$ -structures with quasi-Fuchsian holonomy. (2) The path lifting property of developing maps in the domain of discontinuities in ℂ P 1 $$\mathbb {C}\mathrm {P}^1$$ . Keywords: ℂ P 1 $$\mathbb {C}\mathrm {P}^1$$ -structures; Measured laminations; Pleated surfaces; 57M50 (search for similar items in EconPapers) There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-55928-1_6 Ordering information: This item can be ordered from DOI: 10.1007/978-3-030-55928-1_6 Access Statistics for this chapter More chapters in Springer Books from Springer |
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