A *projective surface* is a pair \((\Sigma,\mathfrak{p})\) consisting of a smooth surface and an equivalence class of torsion-free connections on the tangent bundle.

If the surface is compact, oriented, and of negative Euler-characteristic, then \({\mathfrak p}\) is defined by at most one conformal connection. This result by the author is a rigidity result for the existence of a certain holomorphic curve in a complex surface \(Z\), or equivalently, for the lift of the curve into the projective holomorphic cotangent bundle of \(Z\). One central question of the project is whether this rigidity result still holds true if the curve (or more precisely its lift to the projectivized tangent bundle) is not holomorphic, but merely a minimal surface.

Other questions concern topological properties of projective surfaces. For example, one might ask whether a compact surface with \(\pm\chi(\Sigma)>0\) can carry a torsion-free connection whose Ricci curvature is symmetric and negative / positive definite.

## Publications

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)

## Team Members

**JProf. Dr. Thomas Mettler**

Project leader

Goethe-Universität Frankfurt

mettler(at)math.uni-frankfurt.de