Convex projective surfaces with compatible Weyl connection are hyperbolic

DOI10.2140/APDE.2020.13.1073arXiv1804.04616MaRDI QIDQ6300283

Thomas Mettler, Gabriel P. Paternain

Publication date: 12 April 2018

Abstract: We show that a properly convex projective structure $mathfrakp$ on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if $mathfrakp$ is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that $mathfrakp$ admits a compatible Weyl connection if and only if a certain holomorphic curve exists. Turning this non-linear 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.

Mathematics Subject Classification ID

Projective differential geometry (53A20) Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) (37D40) Differentials on Riemann surfaces (30F30) Other partial differential equations of complex analysis in several variables (32W50)

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