Dividing sets as nodal sets of an eigenfunction of the Laplacian

Abstract

We show that for any convex surface $S$ in a contact $3$–manifold, there exists a metric on $S$ and a neighbourhood contact isotopic to $S\times I$ with the contact structure given by $ker\left(udt-\star du\right)$ where $u$ is an eigenfunction of the Laplacian on $S$ and $\star$ is the Hodge star from the metric on $S$. This answers a question posed by Komendarczyk [Trans. Amer. Math. Soc. 358 (2006) 2399–2413].