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Convex projective
surfaces with compatible Weyl connection are hyperbolic
Thomas Mettler and Gabriel P. Paternain |
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Vol. 13 (2020), No. 4, 1073–1097
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Abstract |
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We show that a properly convex projective structure on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if is hyperbolic. We phrase the problem as a nonlinear PDE for a Beltrami differential by using that 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 -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
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Mathematical Subject Classification 2010
Primary: 32W50, 53A20
Secondary: 30F30, 37D40
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Milestones
Received: 14 June 2018
Revised: 20 April 2019
Accepted: 1 June 2019
Published: 13 June 2020
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Authors |
| Thomas Mettler | |
| Institut für Mathematik Goethe-Universität Frankfurt Frankfurt am Main Germany |
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| Gabriel P. Paternain | |
| Department of Pure Mathematics and
Mathematical Statistics University of Cambridge Cambridge United Kingdom |
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