Convex projective surfaces with compatible Weyl connection are hyperbolic

Mettler, Thomas; Paternain, Gabriel P.

doi:10.2140/apde.2020.13.1073

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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

DOI: 10.2140/apde.2020.13.1073

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 $L^{2}$ -energy identity known as Pestov’s identity to prove a vanishing theorem for the relevant transport equation.

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

Vol. 13, No. 4, 2020