## Convex projective surfaces with compatible Weyl connection are hyperbolic. (arXiv:1804.04616v1 [math.DG])

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. 查看全文>>