Cet article concerne le nombre de domaines nodaux des fonctions propres du Laplacien sur des variétés Riemanniennes Kaluza–Klein en dimension trois, à savoir des variétés qui sont des fibrés -principaux sur des surfaces de Riemann équipées avec une métrique -invariante de type Kaluza–Klein. Pour des métriques génériques de ce type, on prouve que chaque fonction propre possède exactement deux domains nodaux, sauf si elle est invariante par l’action de .
On construit aussi une base orthonormale de fonctions propres explicites du tore plat pour que chaque fonction propre non constante possède exactement deux domaines nodaux.
This article concerns the number of nodal domains of eigenfunctions of the Laplacian on special Riemannian -manifolds, namely nontrivial principal bundles over Riemann surfaces equipped with certain invariant metrics, the Kaluza–Klein metrics. We prove for generic Kaluza–Klein metrics that any Laplacian eigenfunction has exactly two nodal domains unless it is invariant under the action.
We also construct an explicit orthonormal eigenbasis on the flat -torus for which every non-constant eigenfunction has two nodal domains.
Révisé le : 2019-07-15
Accepté le : 2019-09-18
Publié le : 2020-12-18
Classification : 58J50
Mots clés : fonction propre du Laplacien, fibré principal, métrique de Kaluza–Klein, domaine nodal
@article{AIF_2020__70_3_971_0, author = {Jung, Junehyuk and Zelditch, Steve}, title = {Boundedness of the number of nodal domains for eigenfunctions of generic Kaluza--Klein 3-folds}, journal = {Annales de l'Institut Fourier}, pages = {971--1027}, publisher = {Association des Annales de l'institut Fourier}, volume = {70}, number = {3}, year = {2020}, doi = {10.5802/aif.3329}, language = {en}, url = {https://aif.centre-mersenne.org/item/AIF_2020__70_3_971_0/} }
Jung, Junehyuk; Zelditch, Steve. Boundedness of the number of nodal domains for eigenfunctions of generic Kaluza–Klein 3-folds. Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 971-1027. doi : 10.5802/aif.3329. https://aif.centre-mersenne.org/item/AIF_2020__70_3_971_0/
[1] A generalization of the Gauss-Bonnet-Hopf-Poincaré formula for sections and branched sections of bundles, J. Geom. Phys., Volume 121 (2017), pp. 108-122 | Article | MR 3705384 | Zbl 1375.53036
[2] Experimental Nazarov-Sodin constants, genus, and percolation on nodal domains for D and D random waves (2017) (in preparation)
[3] Laplacians and Riemannian submersions with totally geodesic fibres, Ill. J. Math., Volume 26 (1982) no. 2, pp. 181-200 | MR 650387
[4] Methods of mathematical physics. Vol. I, Interscience Publishers, 1953, xv+561 pages | MR 0065391
[5] Analytic methods in algebraic geometry, Surveys of Modern Mathematics, Volume 1, International Press; Higher Education Press, 2012, viii+231 pages | MR 2978333 | Zbl 1271.14001
[6] Nondegeneracy of the eigenvalues of the Hodge Laplacian for generic metrics on 3-manifolds, Trans. Am. Math. Soc., Volume 364 (2012) no. 8, pp. 4207-4224 | Article | MR 2912451
[7] Isolated singularities of binary differential equations of degree , Publ. Mat., Barc., Volume 56 (2012) no. 1, pp. 65-89 | Article | MR 2918184 | Zbl 1282.37017
[8] Nodal domains of Maass forms I, Geom. Funct. Anal., Volume 23 (2013) no. 5, pp. 1515-1568 | Article | MR 3102912
[9] Nodal domains of Maass forms, II, Am. J. Math., Volume 139 (2017) no. 5, pp. 1395-1447 | Article | MR 3702502
[10] Analytic number theory, Colloquium Publications, Volume 53, American Mathematical Society, 2004, xii+615 pages | Article | MR 2061214
[11] Quantum unique ergodicity and the number of nodal domains of eigenfunctions, J. Am. Math. Soc., Volume 31 (2018) no. 2, pp. 303-318 | Article | MR 3758146
[12] Sign changes of the Eisenstein series on the critical line, Int. Math. Res. Not. (2019) no. 3, pp. 641-672 | Article | MR 3910468
[13] Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution, J. Differ. Geom., Volume 102 (2016) no. 1, pp. 37-66 | MR 3447086
[14] Number of nodal domains of eigenfunctions on non-positively curved surfaces with concave boundary, Math. Ann., Volume 364 (2016) no. 3-4, pp. 813-840 | Article | MR 3466853
[15] Topology of the nodal set of random equivariant spherical harmonics on (2019) (https://arxiv.org/abs/1908.00979, to appear in Int. Math. Res. Not.)
[16] On the minimum number of domains in which the nodal lines of spherical harmonics divide the sphere, Commun. Partial Differ. Equations, Volume 2 (1977) no. 12, pp. 1233-1244 | MR 0477199 | Zbl 0377.31008
[17] Arithmetic, zeros, and nodal domains on the sphere, Commun. Math. Phys., Volume 338 (2015) no. 3, pp. 919-951 | Article | MR 3355806
[18] Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen. Math.- naturwiss. Diss., Göttingen, 30 S (1925)., 1925 | Zbl 51.0356.01
[19] Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Volume 5, Springer, 1984, xii+184 pages | Article | MR 743423
[20] Generic properties of eigenfunctions, Am. J. Math., Volume 98 (1976) no. 4, pp. 1059-1078 | Article | MR 0464332 | Zbl 0355.58017
[21] Totally geodesic maps, J. Differ. Geom., Volume 4 (1970), pp. 73-79 | MR 0262984
[22] Logarithmic lower bound on the number of nodal domains, J. Spectr. Theory, Volume 6 (2016) no. 4, pp. 1047-1086 | Article | MR 3584194