Research interests

I am mainly concerned with using Riemannian geometry to understand contact structures. Occasionally it turns out that the other direction is of interest as well, which is something that always makes me happy.

The foundation of this unlikely confluence is the concept of (weak) compatibility between a Riemannian metric gg and a contact form α\alpha. Such a pair (g,α)(g,\alpha) is called weakly compatible if gdα=λα\star_g d\alpha=\lambda \alpha for λR{0}\lambda\in \mathbb{R}\setminus \{0\} (sometimes the condition is weakened to allow for λ\lambda to be a positive or negative function) and compatible if furthermore αg=const\lVert \alpha\rVert_g=const.

The other direction, i.e. finding a (weakly) compatible contact form given a Riemannian metric, is not as straighforward. The idea is to find solutions to the above first-order linear PDE and find a way to ensure that it is nowhere 00. It is worth mentioning that solutions to this equation are of interest in both geometry (as the operator gd\star_g d really is the square root of the Hodge Laplacian on coclosed 11-forms) and fluid dynamics (solutions to gdα=λα\star_g d\alpha=\lambda \alpha happen to satisfy Euler's equation). Using tools from contact topology and the notion of weak compatibility it is possible to construct solutions with remarkable properties (e.g. ones that have complicated closed orbits) or prove that the set of Riemannian metrics that admit nowhere zero solutions to Euler's equations is L2L^2-dense in the space of Riemannian metrics for any closed 33-manifold.

There are several conditions on a (weakly) compatible metric that ensures that corresponding contact forms define tight contact structures. The hope of this approach to contact topology is to prove the existence of tight contact structures on various classes of 33-manifolds, most notably hyperbolic ones. I found new such criteria (still unpublished, write me an email if this interests you), and believe that understanding metrics for which the kk-th eigenvalue functional is critical might be helpful for making progress on this problem.

Apart from this intersection of contact topology and Riemannian geometry I am interested in studying the properties of the operator gdα\star_g d\alpha more generally. This lead to further interest in spectral theory, and ultimately culminated in the resolution of Arnold's transversality conjecture for the Laplace-Beltrami operator (joint work with Josef Greilhuber).

Currently I am trying to use ideas from minimal surface theory in order to restrain the topology of pseudoholomorphic curves in symplectizations of positively curves contact manifolds.

Otherwise I am interested in all things low dimensional topology and geometry and I am always happy to discuss math with people, so feel free to talk to me whenever you see me 😃