On the contact geometry of nodal sets

Komendarczyk, R.

doi:10.1090/S0002-9947-05-03970-X

On the contact geometry of nodal sets
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Trans. Amer. Math. Soc. 358 (2006), 2399-2413 Request permission

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

Giovanni Alessandrini, Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains, Comment. Math. Helv. 69 (1994), no. 1, 142–154. MR 1259610, DOI 10.1007/BF02564478
Colette Anné, Perturbation du spectre $X\sbs TUB^\epsilon Y$ (conditions de Neumann), Séminaire de Théorie Spectrale et Géométrie, No. 4, Année 1985–1986, Univ. Grenoble I, Saint-Martin-d’Hères, 1986, pp. 17–23 (French). MR 1046060
Colette Anné and Bruno Colbois, Opérateur de Hodge-Laplace sur des variétés compactes privées d’un nombre fini de boules, J. Funct. Anal. 115 (1993), no. 1, 190–211 (French, with English summary). MR 1228148, DOI 10.1006/jfan.1993.1087
Thierry Aubin, Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1636569, DOI 10.1007/978-3-662-13006-3
Shigetoshi Bando and Hajime Urakawa, Generic properties of the eigenvalue of the Laplacian for compact Riemannian manifolds, Tohoku Math. J. (2) 35 (1983), no. 2, 155–172. MR 699924, DOI 10.2748/tmj/1178229047
Shiu Yuen Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), no. 1, 43–55. MR 397805, DOI 10.1007/BF02568142
S. S. Chern and R. S. Hamilton, On Riemannian metrics adapted to three-dimensional contact manifolds, Workshop Bonn 1984 (Bonn, 1984) Lecture Notes in Math., vol. 1111, Springer, Berlin, 1985, pp. 279–308. With an appendix by Alan Weinstein. MR 797427, DOI 10.1007/BFb0084596
Y. Eliashberg, Classification of overtwisted contact structures on $3$-manifolds, Invent. Math. 98 (1989), no. 3, 623–637. MR 1022310, DOI 10.1007/BF01393840
John B. Etnyre, Tight contact structures on lens spaces, Commun. Contemp. Math. 2 (2000), no. 4, 559–577. MR 1806947, DOI 10.1142/S0219199700000207
John Etnyre and Robert Ghrist, Contact topology and hydrodynamics. I. Beltrami fields and the Seifert conjecture, Nonlinearity 13 (2000), no. 2, 441–458. MR 1735969, DOI 10.1088/0951-7715/13/2/306
Pedro Freitas, Closed nodal lines and interior hot spots of the second eigenfunction of the Laplacian on surfaces, Indiana Univ. Math. J. 51 (2002), no. 2, 305–316. MR 1909291, DOI 10.1512/iumj.2002.51.2208
David B. Gauld, Differential topology, Monographs and Textbooks in Pure and Applied Mathematics, vol. 72, Marcel Dekker, Inc., New York, 1982. An introduction. MR 680937
R. Ghrist and R. Komendarczyk. Overtwisted energy-minimizing curl eigenfields. preprint, arXiv:math.SG/0411319.
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Emmanuel Giroux, Structures de contact sur les variétés fibrées en cercles audessus d’une surface, Comment. Math. Helv. 76 (2001), no. 2, 218–262 (French, with French summary). MR 1839346, DOI 10.1007/PL00000378
M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and N. Nadirashvili, On the nodal line conjecture, Advances in differential equations and mathematical physics (Atlanta, GA, 1997) Contemp. Math., vol. 217, Amer. Math. Soc., Providence, RI, 1998, pp. 33–48. MR 1605269, DOI 10.1090/conm/217/02980
Ko Honda, On the classification of tight contact structures. I, Geom. Topol. 4 (2000), 309–368. MR 1786111, DOI 10.2140/gt.2000.4.309
Ko Honda, On the classification of tight contact structures. II, J. Differential Geom. 55 (2000), no. 1, 83–143. MR 1849027
Ko Honda, William H. Kazez, and Gordana Matić, Convex decomposition theory, Int. Math. Res. Not. 2 (2002), 55–88. MR 1874319, DOI 10.1155/S1073792802101140
R. Komendarczyk. ————–. Ph.D. thesis – in preparation.
J. Martinet, Formes de contact sur les variétés de dimension $3$, Proceedings of Liverpool Singularities Symposium, II (1969/1970), Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971, pp. 142–163 (French). MR 0350771
Antonios D. Melas, On the nodal line of the second eigenfunction of the Laplacian in $\textbf {R}^2$, J. Differential Geom. 35 (1992), no. 1, 255–263. MR 1152231
L. E. Payne, Isoperimetric inequalities and their applications, SIAM Rev. 9 (1967), 453–488. MR 218975, DOI 10.1137/1009070
Jeffrey Rauch and Michael Taylor, Potential and scattering theory on wildly perturbed domains, J. Functional Analysis 18 (1975), 27–59. MR 377303, DOI 10.1016/0022-1236(75)90028-2
John Roe, Elliptic operators, topology and asymptotic methods, 2nd ed., Pitman Research Notes in Mathematics Series, vol. 395, Longman, Harlow, 1998. MR 1670907
Steven Rosenberg, The Laplacian on a Riemannian manifold, London Mathematical Society Student Texts, vol. 31, Cambridge University Press, Cambridge, 1997. An introduction to analysis on manifolds. MR 1462892, DOI 10.1017/CBO9780511623783
Junya Takahashi, Collapsing of connected sums and the eigenvalues of the Laplacian, J. Geom. Phys. 40 (2002), no. 3-4, 201–208. MR 1866987, DOI 10.1016/S0393-0440(01)00033-X
K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math. 98 (1976), no. 4, 1059–1078. MR 464332, DOI 10.2307/2374041