Polynomial cubic differentials and convex polygons in the projective\n plane

2014/07/30 by David Dumas, Michael Wolf, Dumas, David +2 · 2 citations
Mathematics · #30F60 #53A15 #Advanced Differential Equations and Dynamical Systems #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1407.8149

openalex publication_date 2014/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct and study a natural homeomorphism between the moduli space of\npolynomial cubic differentials of degree d on the complex plane and the space\nof projective equivalence classes of oriented convex polygons with d+3\nvertices. This map arises from the construction of a complete hyperbolic affine\nsphere with prescribed Pick differential, and can be seen as an analogue of the\nLabourie-Loftin parameterization of convex RP² structures on a compact surface\nby the bundle of holomorphic cubic differentials over Teichmuller space.\n

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