Polynomial cubic differentials and convex polygons in the projective plane

By David Dumas, Michael Wolf. 2014

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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 $${{\mathbb{RP}}^2}$$RP2 structures on a compact surface by the bundle of holomorphic cubic differentials Teichmüller space.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d4edb3fe53bc2f7f04ac6a9f7f214690286d1d6c