Handbook of Teichmüller Theory, Volume II

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pp: 455–508

DOI: 10.4171/055-1/13

Complex projective structures

David Dumas[1]

(1) University of Illinois at Chicago, USA

This is a survey of the theory of complex projective ℂℙ1-structures on compact surfaces. After some preliminary discussion and definitions, we concentrate on three main topics:

  1. Using the Schwarzian derivative to parameterize the moduli space
  2. Thurston’s parameterization of the moduli space using grafting
  3. Holonomy representations of ℂℙ1-structures
We also discuss some results comparing the two parameterizations of the space of projective structures and relating these parameterizations to the holonomy map.

Keywords: Complex projective structure, Schwarzian derivative, grafting, bending, Thurston metric, holonomy, earthquake, quakebend, collapsing map, harmonic map