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Projective surfaces, Segre structures and the Hitchin component for PSL(n,R)

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 the metrisability of an oriented projective surface is equivalent to the existence of pseudo-holomorphic curves. A projective structure $\mathfrak{p}$ and a volume form $\sigma$ on an oriented surface $M$ equip the total space of a certain disk bundle $Z \to M$ with a pair $(J_{\mathfrak{p}},\mathfrak{J}_{\mathfrak{p},\sigma})$ of almost complex structures. A conformal structure on $M$ corresponds to a section of $Z\to M$ and $\mathfrak{p}$ is metrisable by the metric $g$ if and only if $[g] : M \to Z$ is a pseudo-holomorphic curve with respect to $J_{\mathfrak{p}}$ and $\mathfrak{J}_{\mathfrak{p},dA_g}$.

 Journal Mathematische Zeitschrift Publisher Springer Link to preprint version Link to published version

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.

 Journal Analysis & PDE Publisher Mathematical Sciences Publishers Volume 13 Pages 1073--1097 Link to preprint version Link to published version

We introduce a new family of thermostat flows on the unit tangent bundle of an oriented Riemannian 2-manifold. Suitably reparametrised, these flows include the geodesic flow of metrics of negative Gauss curvature and the geodesic flow induced by the Hilbert metric on the quotient surface of divisible convex sets. We show that the family of flows can be parametrised in terms of certain weighted holomorphic differentials and investigate their properties. In particular, we prove that they admit a dominated splitting and we identify special cases in which the flows are Anosov. In the latter case, we study when they admit an invariant measure in the Lebesgue class and the regularity of the weak foliations.

 Journal Mathematische Annalen Publisher Springer Volume 373 Pages 553--580 Link to preprint version Link to published version
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## Former Members

Lukas Poerschke
Doctoral student
Goethe Universität Frankfurt