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Polynomial cubic differentials and convex polygons in the projective plane


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.

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Dumas, D., Wolf, M. Polynomial cubic differentials and convex polygons in the projective plane. Geom. Funct. Anal. 25, 1734–1798 (2015).

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  • Modulus Space
  • Riemann Surface
  • Convex Polygon
  • Convex Domain
  • Projective Transformation