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2020 Convex projective surfaces with compatible Weyl connection are hyperbolic
Thomas Mettler, Gabriel P. Paternain
Anal. PDE 13(4): 1073-1097 (2020). 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 L2-energy identity known as Pestov’s identity to prove a vanishing theorem for the relevant transport equation.

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Thomas Mettler. Gabriel P. Paternain. "Convex projective surfaces with compatible Weyl connection are hyperbolic." Anal. PDE 13 (4) 1073 - 1097, 2020. https://doi.org/10.2140/apde.2020.13.1073

Information

Received: 14 June 2018; Revised: 20 April 2019; Accepted: 1 June 2019; Published: 2020
First available in Project Euclid: 25 June 2020

zbMATH: 07221197
MathSciNet: MR4109900
Digital Object Identifier: 10.2140/apde.2020.13.1073

Subjects:
Primary: 32W50, 53A20
Secondary: 30F30, 37D40

Rights: Copyright © 2020 Mathematical Sciences Publishers

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