Geometric and Functional Analysis

, Volume 25, Issue 6, pp 1734–1798 | Cite as

Polynomial cubic differentials and convex polygons in the projective plane

  • David Dumas
  • Michael Wolf


We construct and study a natural homeomorphism between the moduli space of polynomial cubic differentials of degree d on the complex plane and the space of projective equivalence classes of oriented convex polygons with d + 3 vertices. This map arises from the construction of a complete hyperbolic affine sphere with prescribed Pick differential, and can be seen as an analogue of the Labourie–Loftin parameterization of convex \({{\mathbb{RP}}^2}\) structures on a compact surface by the bundle of holomorphic cubic differentials Teichmüller space.


Modulus Space Riemann Surface Convex Polygon Convex Domain Projective Transformation 
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  1. AS72.
    M. Abramowitz and I. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1972).Google Scholar
  2. BB04.
    O. Biquard and P. Boalch. Wild non-abelian Hodge theory on curves. Compositio Mathematica, (1)140 (2006), 179–204.Google Scholar
  3. Ber09.
    Bernig A.: Hilbert geometry of polytopes. Archiv der Mathematik (Basel) 92(4), 314–324 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  4. BF42.
    Ch. Blanc and F. Fiala. Le type d’une surface et sa courbure totale. Commentarii Mathematici Helvetici, 14 (1942), 230–233.Google Scholar
  5. BH13.
    Benoist Y., Hulin D.: Cubic differentials and finite volume convex projective surfaces. Geometry and Topology 17(1), 595–620 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  6. BH14.
    Y. Benoist and D. Hulin. Cubic differentials and hyperbolic convex sets. Journal of Differential Geometry, (1)98 (2014), 1–19.Google Scholar
  7. BK53.
    H. Busemann and P. Kelly. Projective Geometry and Projective Metrics. Academic Press Inc, New York (1953).Google Scholar
  8. Bla23.
    W. Blaschke. Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. II. Affine Differentialgeometrie, bearbeitet von K. Reidemeister. Erste und zweite Auflage. Springer, Berlin, (1923).Google Scholar
  9. BMS12.
    F. Bonsante, G. Mondello, and J.-M. Schlenker. A cyclic extension of the earthquake flow II (2012) (preprint). arXiv:1208.1738
  10. BMS13.
    F. Bonsante, G. Mondello, and J.-M. Schlenker. A cyclic extension of the earthquake flow I. Geometry and Topology, (1)17(2013), 157–234.Google Scholar
  11. Boa14.
    Boalch P.: Geometry and braiding of Stokes data; fission and wild character varieties. Annals of Mathematics (2) 179(1), 301–365 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  12. Bra91.
    S.B. Bradlow. Special metrics and stability for holomorphic bundles with global sections. Journal of Differential Geometry, (1)33 (1991), 169–213.Google Scholar
  13. Cal72.
    E. Calabi. Complete affine hyperspheres. I. In: Symposia Mathematica, Vol. X (Convegno di Geometria Differenziale, INDAM, Rome, 1971). Academic Press, London (1972), pp. 19–38.Google Scholar
  14. CV11.
    Colbois B., Verovic P.: Hilbert domains that admit a quasi-isometric embedding into Euclidean space. Advances in Geometry 11(3), 465–470 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  15. CVV11.
    B. Colbois, C. Vernicos, and P. Verovic. Hilbert geometry for convex polygonal domains. Journal of Geometry, (1–2)100 (2011), 37–64.Google Scholar
  16. CY75.
    S.Y. Cheng and S.-T. Yau. Differential equations on Riemannian manifolds and their geometric applications. Communications on Pure and Applied Mathematics, (3)28 (1975), 333–354.Google Scholar
  17. CY86.
    Cheng S.Y., Yau S.-T.: Complete affine hypersurfaces. I. The completeness of affine metrics. Communications on Pure and Applied Mathematics 39(6), 839–866 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  18. dF04.
    T. de Fernex. On planar Cremona maps of prime order. Nagoya Mathematical Journal, 174 (2004), 1–28.Google Scholar
  19. dlH93.
    P. de la Harpe. On Hilbert’s metric for simplices. In: Geometric Group Theory, Vol. 1 (Sussex, 1991). London Mathematical Society Lecture Note Series, Vol. 181. Cambridge University Press, Cambridge (1993), pp. 97–119.Google Scholar
  20. Dun12.
    Dunajski M.: Abelian vortices from sinh-Gordon and Tzitzeica equations. Physics Letters B 710(1), 236–239 (2012)CrossRefMathSciNetGoogle Scholar
  21. FG07.
    V.V. Fock and A.B. Goncharov. Moduli spaces of convex projective structures on surfaces. Advances in Mathematics, (1)208 (2007), 249–273.Google Scholar
  22. Fin65.
    R. Finn. On a class of conformal metrics, with application to differential geometry in the large. Commentarii Mathematici Helvetici, 40 (1965), 1–30.Google Scholar
  23. FK05.
    T. Foertsch and A. Karlsson. Hilbert metrics and Minkowski norms. Journal of Geometry, (1–2)83 (2005), 22–31.Google Scholar
  24. Gig81.
    Gigena S.: On a conjecture by E Calabi.. Geometriae Dedicata 11(4), 387–396 (1981)CrossRefMathSciNetzbMATHGoogle Scholar
  25. GL50.
    V.L. Ginburg and L.D. Landau. Zh. Eksp. Teor. Fiz., 20 (1950), 1064. English translation in Collected Papers of L.D. Landau, pp. 546–568, Pergamon Press (1965).Google Scholar
  26. GP94.
    O. García-Prada. A direct existence proof for the vortex equations over a compact Riemann surface. Bulletin of the London Mathematical Society, (1)26 (1994), 88–96.Google Scholar
  27. GT83.
    D. Gilbarg and N.S. Trudinger. Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften, Vol. 224, 2nd edn. Springer, Berlin (1983).Google Scholar
  28. Han96.
    Z.-C. Han. Remarks on the geometric behavior of harmonic maps between surfaces. In: Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994). A K Peters, Wellesley (1996), pp. 57–66.Google Scholar
  29. Har02.
    P. Hartman. Ordinary Differential Equations. Classics in Applied Mathematics, Vol. 38. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002)Google Scholar
  30. HTTW95.
    Z.-C. Han, L.-F. Tam, A. Treibergs, and T. Wan. Harmonic maps from the complex plane into surfaces with nonpositive curvature. Communications in Analysis and Geometry, (1–2)3 (1995), 85–114.Google Scholar
  31. Hub57.
    Huber A.: On subharmonic functions and differential geometry in the large. Commentarii Mathematici Helvetici 32, 13–72 (1957)CrossRefMathSciNetzbMATHGoogle Scholar
  32. Jos07.
    J. Jost. Partial Differential Equations. Graduate Texts in Mathematics, Vol. 214, 2nd edn. Springer, New York (2007).Google Scholar
  33. JT80.
    A. Jaffe and C. Taubes. Vortices and Monopoles. Progress in Physics, Vol. 2. Birkhäuser, Boston (1980). Structure of static gauge theories.Google Scholar
  34. Kay67.
    Kay D.: The ptolemaic inequality in Hilbert geometries.. Pacific Journal of Mathematics 21, 293–301 (1967)CrossRefMathSciNetzbMATHGoogle Scholar
  35. Lab07.
    F. Labourie. Flat projective structures on surfaces and cubic holomorphic differentials. Pure and Applied Mathematics Quarterly (4, part 1)3 (2007), 1057–1099.Google Scholar
  36. Li90.
    A.M. Li. Calabi conjecture on hyperbolic affine hyperspheres. Mathematische Zeitschrift, (3)203 (1990), 483–491.Google Scholar
  37. Li92.
    Li A.M.: Calabi conjecture on hyperbolic affine hyperspheres. II.. Mathematische Annalen 293(3), 485–493 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  38. LLS04.
    A.-M. Li, H. Li, and U. Simon. Centroaffine Bernstein problems. Differential Geometry and its Applications, (3)20 (2004), 331–356.Google Scholar
  39. Lof01.
    J. Loftin. Affine spheres and convex \({{\mathbb{RP}}^n}\)-manifolds. American Journal of Mathematics, (2)123 (2001), 255–274.Google Scholar
  40. Lof04.
    J. Loftin. The compactification of the moduli space of convex \({{\mathbb{RP}}^2}\) surfaces. I. Journal of Differential Geometry, (2)68 (2004), 223–276.Google Scholar
  41. Lof07.
    Loftin J.: Flat metrics, cubic differentials and limits of projective holonomies. Geometriae Dedicata 128, 97–106 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  42. Lof10.
    J. Loftin. Survey on affine spheres. In: Handbook of Geometric Analysis, No. 2. Advanced Lectures in Mathematics, Vol. 13. International Press, Somerville (2010), pp. 161–191.Google Scholar
  43. Lof15.
    J. Loftin. Convex \({{\mathbb{RP}}^2}\) structures and cubic differentials under neck separation (2015) (preprint). arXiv:1506.03895
  44. LSZ93.
    A.M. Li, U. Simon, and G.S. Zhao. Global Affine Differential Geometry of Hypersurfaces. de Gruyter Expositions in Mathematics, Vol. 11. Walter de Gruyter & Co, Berlin (1993).Google Scholar
  45. Min92.
    Minsky Y.: Harmonic maps, length, and energy in Teichmüller space. Journal of Differential Geometry 35(1), 151–217 (1992)MathSciNetzbMATHGoogle Scholar
  46. NS94.
    K. Nomizu and T. Sasaki. Affine Differential Geometry. Cambridge Tracts in Mathematics, Vol. 111. Cambridge University Press, Cambridge (1994).Google Scholar
  47. Omo67.
    H. Omori. Isometric immersions of Riemannian manifolds. Journal of the Mathematical Society of Japan, 19 (1967), 205–214.Google Scholar
  48. Oss86.
    R. Osserman. A Survey of Minimal Surfaces, 2nd edn. Dover Publications Inc, New York (1986).Google Scholar
  49. OST10.
    V. Ovsienko, R. Schwartz, and S. Tabachnikov. The pentagram map: a discrete integrable system. Communications in Mathematical Physics, (2)299 (2010), 409–446.Google Scholar
  50. Pic17.
    Pick G.: Über affine Geometrie iv: Differentialinvarianten der Flächen gegenüber affinen Transformationen. Leipziger Berichte 69, 107–136 (1917)Google Scholar
  51. PZ02.
    A.D. Polyanin and V.F. Zaitsev. Handbook of Linear Partial Differential Equations for Engineers and Scientists. CRC Press, Boca Raton (2002).Google Scholar
  52. Sas80.
    T. Sasaki. Hyperbolic affine hyperspheres. Nagoya Mathematical Journal, 77 (1980), 107–123.Google Scholar
  53. Sch92.
    R. Schwartz. The pentagram map. Experimental Mathematics, (1)1 (1992), 71–81.Google Scholar
  54. Sim90.
    Simpson C.: Harmonic bundles on noncompact curves.. Journal of the American Mathematical Society 3(3), 713–770 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  55. Str84.
    K. Strebel. Quadratic Differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 5. Springer, Berlin (1984).Google Scholar
  56. SW93.
    U. Simon and C.P. Wang. Local theory of affine 2-spheres. In: Differential Geometry: Riemannian Geometry (Los Angeles, CA, 1990). Proceedings of Symposia in Pure Mathematics, Vol. 54. American Mathematical Society, Providence (1993), pp. 585–598.Google Scholar
  57. TW02.
    Trudinger N.S., Wang X.-J.: Affine complete locally convex hypersurfaces. Inventiones Mathematicae 150(1), 45–60 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  58. Tzi08.
    G. Tzitzéica. Sur une nouvelle classe de surfaces. Rendiconti del Circolo Matematico di Palermo, (1)25 (1908), 180–187.Google Scholar
  59. WA94.
    T.Y. Wan and T.K. Au. Parabolic constant mean curvature spacelike surfaces. Proceedings of the American Mathematical Society, (2)120 (1994), 559–564.Google Scholar
  60. Wan91.
    C.P. Wang. Some examples of complete hyperbolic affine 2-spheres in R 3. In: Global Differential Geometry and Global Analysis (Berlin, 1990). Lecture Notes in Mathematics, Vol. 1481. Springer, Berlin, (1991), pp. 271–280.Google Scholar
  61. Wan92.
    Wan T.Y.: Constant mean curvature surface, harmonic maps, and universal Teichmüller space. Journal of Differential Geometry 35(3), 643–657 (1992)MathSciNetzbMATHGoogle Scholar
  62. Wit07.
    E. Witten. From superconductors and four-manifolds to weak interactions. Bulletin of the American Mathematical Society (N.S.), (3)44 (2007), 361–391 (electronic).Google Scholar
  63. Wit08.
    E. Witten. Gauge theory and wild ramification. Analysis and Applications (Singapore), (4)6 (2008), 429–501.Google Scholar
  64. Wol91.
    M. Wolf. High energy degeneration of harmonic maps between surfaces and rays in Teichmüller space. Topology, (4)30 (1991), 517–540.Google Scholar
  65. Yau75.
    Yau S.T.: Harmonic functions on complete Riemannian manifolds.. Communications on Pure and Applied Mathematics 28, 201–228 (1975)CrossRefMathSciNetzbMATHGoogle Scholar

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© Springer International Publishing 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  2. 2.Department of MathematicsRice UniversityHoustonUSA

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