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The classification of real projective structures on compact surfaces


Authors: Suhyoung Choi and William M. Goldman
Journal: Bull. Amer. Math. Soc. 34 (1997), 161-171
MSC (1991): Primary 57M05, 53A20
DOI: https://doi.org/10.1090/S0273-0979-97-00711-8
MathSciNet review: 1414974
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Abstract: Real projective structures (${\Bbb {RP}}^2$-structures) on compact surfaces are classified. The space of projective equivalence classes of real projective structures on a closed orientable surface of genus $g>1$ is a countable disjoint union of open cells of dimension $16g-16$. A key idea is Choi's admissible decomposition of a real projective structure into convex subsurfaces along closed geodesics. The deformation space of convex structures forms a connected component in the moduli space of representations of the fundamental group in $\bold {PGL}(3,{\Bbb R})$, establishing a conjecture of Hitchin.


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Additional Information

Suhyoung Choi
Affiliation: Department of Mathematics, College of Natural Sciences, Seoul National University, 151-742 Seoul, Korea
Email: shchoi@math.snu.ac.kr

William M. Goldman
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: wmg@math.umd.edu

DOI: https://doi.org/10.1090/S0273-0979-97-00711-8
Keywords: Real projective structure, convex real projective structure, deformation space, representation of fundamental groups, developing map, holonomy, Teichmüller, Higgs bundle, $\bf{SL}(3,\Bbb{R})$-representation variety
Received by editor(s): April 15, 1994
Received by editor(s) in revised form: October 13, 1996
Additional Notes: Choi gratefully acknowledges partial support from GARC-KOSEF
Goldman gratefully acknowledges partial support from the National Science Foundation, the Alfred P. Sloan Foundation and the Institute for Advanced Computer Studies at the University of Maryland.
Article copyright: © Copyright 1997 American Mathematical Society

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