Geometric and Functional Analysis

, Volume 25, Issue 6, pp 1734–1798

Polynomial cubic differentials and convex polygons in the projective plane


DOI: 10.1007/s00039-015-0344-5

Cite this article as:
Dumas, D. & Wolf, M. Geom. Funct. Anal. (2015) 25: 1734. doi:10.1007/s00039-015-0344-5


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.

Copyright information

© 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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