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 nonlinear 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 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.