Dividing sets as nodal sets of an eigenfunction of the Laplacian

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 $ker (u d t - ⋆ d u)$ where $u$ is an eigenfunction of the Laplacian on $S$ and $⋆$ is the Hodge star from the metric on $S$ . This answers a question posed by Komendarczyk [Trans. Amer. Math. Soc. 358 (2006) 2399–2413].

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Keywords

contact topology, convex surface, dividing set, nodal set, eigenfunction Laplacian

Mathematical Subject Classification 2000

Primary: 57R17

Secondary: 53D10

References

Publication

Received: 1 April 2010

Revised: 3 November 2010

Accepted: 10 January 2011

Published: 17 May 2011

Authors

Samuel T Lisi
Département de Mathématique Université Libre de Bruxelles, CP 218 Boulevard du Triomphe B-1050 Bruxelles Belgium
http://homepages.ulb.ac.be/~samulisi/

Volume 11, issue 3 (2011)

Samuel T Lisi

Abstract