## JProf. Dr. Thomas Mettler

### Project leader

Goethe-Universität Frankfurt

E-mail: mettler(at)math.uni-frankfurt.de

Telephone: +49 69 798-22596

Homepage: http://www.math.uni-frankfurt.de/~mettle...

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.

**Related project(s):****26**Projective surfaces, Segre structures and the Hitchin component for PSL(n,R)