Convex projective surfaces with compatible Weyl connection are hyperbolic

TL;DR

This paper proves that on closed surfaces with negative Euler characteristic, convex projective structures compatible with Weyl connections are precisely the hyperbolic structures, using nonlinear PDEs and geometric inverse problem techniques.

Contribution

It establishes a characterization of hyperbolic structures among convex projective surfaces via Weyl compatibility, employing novel PDE and inverse problem methods.

Findings

Convex projective structures compatible with Weyl connections are hyperbolic.

A nonlinear PDE for Beltrami differentials characterizes the compatibility.

Application of Pestov's identity leads to a vanishing theorem for the transport equation.

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.