Convex projective surfaces with compatible Weyl connection are hyperbolic

Abstract

We show that a properly convex projective structure $\mathfrak{p}$ on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if $\mathfrak{p}$ is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that $\mathfrak{p}$ 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.

arXiv comment23 pages, added Corollary 4.6, references updated, typos corrected

References in corpus (5)

• On Kähler metrisability of two-dimensional complex projective structures
• Weyl metrisability of two-dimensional projective structures
• Holomorphic differentials, thermostats and Anosov flows
• Geodesic rigidity of conformal connections on surfaces
• Minimal Lagrangian connections on compact surfaces

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• The attenuated ray transforms on Gaussian thermostats with negative curvature
• Projectively equivalent Finsler metrics on surfaces of negative Euler characteristic
• Thermostats without conjugate points
• Deformations of the Veronese embedding and Finsler 2-spheres of constant curvature
• Smooth orbit equivalence rigidity for dissipative geodesic flows