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 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 $\mathfrak{J}_{\mathfrak{p},dA_g}$.

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

We associate a flow $\phi$ to a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi$ always admits a dominated splitting and identify special cases in which $\phi$ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$.

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

We establish a one-to-one correspondence between Finsler structures on the \(2\)-sphere with constant curvature \(1\) and all geodesics closed on the one hand, and Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and all of whose geodesics closed on the other hand. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding $$\mathbb{CP}(a_1,a_2)\to \mathbb{CP}(a_1,(a_1+a_2)/2,a_2)$$ of weighted projective spaces provide examples of Finsler \(2\)-spheres of constant curvature and all geodesics closed.

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

This article studies the bundle of Weyl structures associated to a parabolic geometry. Given a parabolic geometry of any type on a smooth manifold \(M\), this is a natural bundle \(A\to M\), whose smooth sections can be identified with Weyl structures for the geometry. We show that the initial parabolic geometry defines a reductive Cartan geometry on \(A\), which induces an almost bi-Lagrangian structure on \(A\) and a compatible linear connection on \(TA\). In the first part of the article, we prove that all elements of the theory of Weyl structures can be interpreted in terms of natural geometric operations on \(A\). In the second part of the article, we turn our point of view around and use the relation to parabolic geometries and Weyl structures to study the intrinsic geometry on \(A\). This geometry is rather exotic outside of the class of torsion free parabolic geometries associated to a \(|1|\)-grading (i.e. AHS structures), to which we restrict for the rest of the article. We prove that the split-signature metric provided by the almost bi-Lagrangian structure is always Einstein. We proceed to study Weyl structures via the submanifold geometry of the image of the corresponding section in \(A\). For Weyl structures satisfying appropriate non-degeneracy conditions, we derive a universal formula for the second fundamental form of this image. In the last part of the article, we show that for locally flat projective structures, this has close relations to solutions of a projectively invariant Monge-Ampère equation and thus to properly convex projective structures. Analogs for other AHS structures are indicated at the end of the article.

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

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 |

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

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.

**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

** Lukas Poerschke**

Doctoral student

Goethe Universität Frankfurt

poerschke(at)math.uni-frankfurt.de