Vol. 13, No. 4, 2020

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Convex projective surfaces with compatible Weyl connection are hyperbolic

Thomas Mettler and Gabriel P. Paternain

Vol. 13 (2020), No. 4, 1073–1097
Abstract

We show that a properly convex projective structure 𝔭 on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if 𝔭 is hyperbolic. We phrase the problem as a nonlinear PDE for a Beltrami differential by using that 𝔭 admits a compatible Weyl connection if and only if a certain holomorphic curve exists. Turning this nonlinear PDE into a transport equation, we obtain our result by applying methods from geometric inverse problems. In particular, we use an extension of a remarkable L2-energy identity known as Pestov’s identity to prove a vanishing theorem for the relevant transport equation.

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Keywords
convex projective structures, Weyl connections, transport equations, energy identity
Mathematical Subject Classification 2010
Primary: 32W50, 53A20
Secondary: 30F30, 37D40
Milestones
Received: 14 June 2018
Revised: 20 April 2019
Accepted: 1 June 2019
Published: 13 June 2020
Authors
Thomas Mettler
Institut für Mathematik
Goethe-Universität Frankfurt
Frankfurt am Main
Germany
Gabriel P. Paternain
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Cambridge
United Kingdom