On the contact geometry of nodal sets
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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.References
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Additional Information
- R. Komendarczyk
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
- Email: rako@math.gatech.edu
- Received by editor(s): March 19, 2004
- Published electronically: December 20, 2005
- Additional Notes: This research was partially supported by NSF grant DMS-0134408
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 2399-2413
- MSC (2000): Primary 53D10
- DOI: https://doi.org/10.1090/S0002-9947-05-03970-X
- MathSciNet review: 2204037