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On the contact geometry of nodal sets


Author: R. Komendarczyk
Journal: Trans. Amer. Math. Soc. 358 (2006), 2399-2413
MSC (2000): Primary 53D10
DOI: https://doi.org/10.1090/S0002-9947-05-03970-X
Published electronically: December 20, 2005
MathSciNet review: 2204037
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Abstract: In the 3-dimensional Riemannian geometry, contact structures equipped with an adapted Riemannian metric are divergence-free, nondegenerate eigenforms of the Laplace-Beltrami operator. We trace out a two-dimensional consequence of this fact: there is a close relationship between the topology of the contact structure on a convex surface in the 3-manifold (the dividing curves) and the nodal curves of Laplacian eigenfunctions on that surface. Motivated by this relationship, we consider a topological version of Payne's conjecture for the free membrane problem. We construct counterexamples to Payne's conjecture for closed Riemannian surfaces. In light of the correspondence between the nodal lines and dividing curves, we interpret the conjecture in terms of the tight versus overtwisted dichotomy for contact structures.


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Additional Information

R. Komendarczyk
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: rako@math.gatech.edu

DOI: https://doi.org/10.1090/S0002-9947-05-03970-X
Keywords: Nodal lines, dividing curves, contact structures, eigenfunctions of Laplacian
Received by editor(s): March 19, 2004
Published electronically: December 20, 2005
Additional Notes: This research was partially supported by NSF grant DMS-0134408
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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