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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.

## Team Members

JProf. Dr. Thomas Mettler