Geometry & Topology

Cubic differentials and finite volume convex projective surfaces

Yves Benoist and Dominique Hulin

Full-text: Open access

Abstract

We prove that there exists a natural bijection between the set of finite volume oriented convex projective surfaces with nonabelian fundamental group and the set of finite volume hyperbolic Riemann surfaces endowed with a holomorphic cubic differential with poles of order at most 2 at the cusps.

Article information

Source
Geom. Topol., Volume 17, Number 1 (2013), 595-620.

Dates
Received: 12 May 2012
Accepted: 10 November 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732533

Digital Object Identifier
doi:10.2140/gt.2013.17.595

Mathematical Reviews number (MathSciNet)
MR3039771

Zentralblatt MATH identifier
1266.30030

Subjects
Primary: 30F30: Differentials on Riemann surfaces 35J96: Elliptic Monge-Ampère equations 53A15: Affine differential geometry 57M50: Geometric structures on low-dimensional manifolds 53C56: Other complex differential geometry [See also 32Cxx]

Keywords
affine spheres convex projective surfaces Teichmüller spaces cubic differentials Monge-Ampère equations

Citation

Benoist, Yves; Hulin, Dominique. Cubic differentials and finite volume convex projective surfaces. Geom. Topol. 17 (2013), no. 1, 595--620. doi:10.2140/gt.2013.17.595. https://projecteuclid.org/euclid.gt/1513732533


Export citation

References

  • L V Ahlfors, An extension of Schwarz's lemma, Trans. Amer. Math. Soc. 43 (1938) 359–364
  • J-P Benzécri, Sur les variétés localement affines et localement projectives, Bull. Soc. Math. France 88 (1960) 229–332
  • E Calabi, Complete affine hyperspheres, I, from: “Symposia Mathematica, Vol. X”, Academic Press, London (1972) 19–38
  • S Y Cheng, S T Yau, On the regularity of the Monge–Ampère equation $\mathrm{ det}(\partial \sp{2}u/\partial x\sb{i}\partial x\sb{j})=F(x,u)$, Comm. Pure Appl. Math. 30 (1977) 41–68
  • S Y Cheng, S T Yau, Complete affine hypersurfaces, I, The completeness of affine metrics, Comm. Pure Appl. Math. 39 (1986) 839–866
  • S Gigena, On a conjecture by E. Calabi, Geom. Dedicata 11 (1981) 387–396
  • D Gilbarg, N S Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer, Berlin (2001)
  • W Goldman, Projective geometry on manifolds, Lecture notes (1988) Available at \setbox0\makeatletter\@url http://www2.math.umd.edu/~wmg/pgom.pdf {\unhbox0
  • W M Goldman, Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990) 791–845
  • F Labourie, Flat projective structures on surfaces and cubic holomorphic differentials, Pure Appl. Math. Q. 3 (2007) 1057–1099
  • J Lee, Convex fundamental domains for properly convex real projective structures Available at \setbox0\makeatletter\@url http://users.math.yale.edu/~jl232/ {\unhbox0
  • A M Li, Calabi conjecture on hyperbolic affine hyperspheres, II, Math. Ann. 293 (1992) 485–493
  • J Loftin, Applications of affine differential geometry to $\mathbb{R}\mathrm{P}^2$ surfaces, PhD thesis, Rutgers University (1999) Available at \setbox0\makeatletter\@url http://andromeda.rutgers.edu/~loftin {\unhbox0
  • J C Loftin, Affine spheres and convex $\mathbb{R}\mathrm{P}\sp n$–manifolds, Amer. J. Math. 123 (2001) 255–274
  • J C Loftin, The compactification of the moduli space of convex $\mathbb{R}\mathrm{P}\sp 2$ surfaces, I, J. Differential Geom. 68 (2004) 223–276
  • J Loftin, Survey on affine spheres, from: “Handbook of geometric analysis, No. 2”, (L Ji, P Li, R Schoen, L Simon, editors), Adv. Lect. Math. (ALM) 13, Int. Press, Somerville, MA (2010) 161–191
  • L Marquis, Espace des modules marqués des surfaces projectives convexes de volume fini, Geom. Topol. 14 (2010) 2103–2149
  • L Marquis, Surface projective convexe de volume fini, Ann. Inst. Fourier (Grenoble) 62 (2012) 325–392
  • K Nomizu, T Sasaki, Affine differential geometry, Cambridge Tracts in Mathematics 111, Cambridge Univ. Press (1994)
  • M Troyanov, The Schwarz lemma for nonpositively curved Riemannian surfaces, Manuscripta Math. 72 (1991) 251–256
  • N S Trudinger, X-J Wang, The Monge–Ampère equation and its geometric applications, from: “Handbook of geometric analysis, No. 1”, (L Ji, P Li, R Schoen, L Simon, editors), Adv. Lect. Math. (ALM) 7, Int. Press, Somerville, MA (2008) 467–524
  • C P Wang, Some examples of complete hyperbolic affine $2$–spheres in $\mathbb{R}\sp 3$, from: “Global differential geometry and global analysis”, (D Ferus, U Pinkall, U Simon, B Wegner, editors), Lecture Notes in Math. 1481, Springer, Berlin (1991) 271–280
  • S T Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975) 201–228