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 $g$ and a contact form $\alpha$. Such a pair $(g,\alpha)$ is called weakly compatible if $\star_g d\alpha=\lambda \alpha$ for $\lambda\in \mathbb{R}\setminus \{0\}$ (sometimes the condition is weakened to allow for $\lambda$ to be a positive or negative function) and compatible if furthermore $\lVert \alpha\rVert_g=const$.

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 $0$. It is worth mentioning that solutions to this equation are of interest in both geometry (as the operator $\star_g d$ really is the square root of the Hodge Laplacian on coclosed $1$-forms) and fluid dynamics (solutions to $\star_g d\alpha=\lambda \alpha$ 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 $L^2$-dense in the space of Riemannian metrics for any closed $3$-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 $3$-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 I found new such criteria (still unpublished, write me an email if this interests you), and believe that understanding metrics for which the $k$-th eigenvalue functional is critical might be helpful for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 $\star_g d\alpha$ 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 😃