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Polynomial cubic differentials and convex polygons in the projective plane

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Abstract

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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  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, USA

    David Dumas

  2. Department of Mathematics, Rice University, Houston, TX, USA

    Michael Wolf

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  1. David Dumas
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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). https://doi.org/10.1007/s00039-015-0344-5

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