Transactions of the American Mathematical Society

Author(s): R. Komendarczyk
Journal: Trans. Amer. Math. Soc. 358 (2006), 2399-2413.
MSC (2000): Primary 53D10
Posted: December 20, 2005
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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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Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53D10

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

DOI: 10.1090/S0002-9947-05-03970-X
PII: S 0002-9947(05)03970-X
Keywords: Nodal lines, dividing curves, contact structures, eigenfunctions of Laplacian
Received by editor(s): March 19, 2004
Posted: December 20, 2005
Additional Notes: This research was partially supported by NSF grant DMS-0134408
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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