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 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.

6 pages, 1 figure

References in corpus (1)

• Tightness in contact metric 3-manifolds

Cited by in corpus (3)

• Contact structures and Beltrami fields on the torus and the sphere
• Constructing Riemannian metrics with prescribed nodal sets for Laplacian eigenfunctions
• Morse functions and contact convex surfaces