Publications

Using the Schwarzian derivative we construct a sequence \(\left(P_{\ell}\right)_{\ell \geqslant 2}\) of meromorphic differentials on every non-flat oriented minimal surface in Euclidean \(3\)-space. The differentials \(\left(P_{\ell}\right)_{\ell \geqslant 2}\) are invariant under all deformations of the surface arising via the Weierstrass representation and depend on the induced metric and its derivatives only. A minimal surface is said to have degree \(n\) if its \(n\)-th differential is a polynomial expression in the differentials of lower degree. We observe that several well-known minimal surfaces have small degree, including Enneper's surface, the helicoid/catenoid and the Scherk -- as well as the Schwarz family. Furthermore, it is shown that locally and away from umbilic points every minimal surface can be approximated by a sequence of minimal surfaces of increasing degree.

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

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):
26Projective 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):
26Projective 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.

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