JProf. Dr. Thomas Mettler
We show that a properly convex projective structure p on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if p is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that 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 L2-energy identity known as Pestov's identity to prove a vanishing theorem for the relevant transport equation.