Boundedness of the number of nodal domains for eigenfunctions of generic Kaluza–Klein 3-folds
Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 971-1027.

This article concerns the number of nodal domains of eigenfunctions of the Laplacian on special Riemannian 3-manifolds, namely nontrivial principal S 1 bundles PX over Riemann surfaces equipped with certain S 1 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 S 1 action.

We also construct an explicit orthonormal eigenbasis on the flat 3-torus 𝕋 3 for which every non-constant eigenfunction has two nodal domains.

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 S 1 -principaux PX sur des surfaces de Riemann équipées avec une métrique S 1 -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 S 1 .

On construit aussi une base orthonormale de fonctions propres explicites du tore plat 𝕋 3 pour que chaque fonction propre non constante possède exactement deux domaines nodaux.

Received: 2018-10-04
Revised: 2019-07-15
Accepted: 2019-09-18
Published online: 2020-06-26
DOI: https://doi.org/10.5802/aif.3329
Classification: 58J50
Keywords: Eigenfunction of the Laplacian, Principal bundle, Kaluza–Klein metric, Nodal domain
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {70},
     number = {3},
     year = {2020},
     pages = {971-1027},
     doi = {10.5802/aif.3329},
     language = {en},
     url = {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, Volume 70 (2020) no. 3, pp. 971-1027. doi : 10.5802/aif.3329. https://aif.centre-mersenne.org/item/AIF_2020__70_3_971_0/

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