Dividing sets as nodal sets of an eigenfunction of the Laplacian

TL;DR

The paper constructs specific contact structures on convex surfaces in 3-manifolds using eigenfunctions of the Laplacian, linking spectral geometry with contact topology.

Contribution

It demonstrates that convex surfaces can be realized as nodal sets of Laplacian eigenfunctions within certain contact structures, answering a previously posed question.

Findings

Existence of metrics making convex surfaces nodal sets of Laplacian eigenfunctions.

Construction of contact structures isotopic to product neighborhoods.

Connection established between spectral geometry and contact topology.

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 contact structure given as $\ker(ud - \star du)$ 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.