no. 3

Boundedness of the number of nodal domains for eigenfunctions of generic Kaluza–Klein 3-folds
[Limite du nombre de domaines nodaux des fonctions propres de variétés Kaluza–Klein génériques en dimension 3]
Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 971-1027.

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 P→X 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.

This article concerns the number of nodal domains of eigenfunctions of the Laplacian on special Riemannian 3-manifolds, namely nontrivial principal S 1 bundles P→X 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.

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DOI : https://doi.org/10.5802/aif.3329
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{\textendash}Klein 3-folds},
     journal = {Annales de l'Institut Fourier},
     pages = {971--1027},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
     number = {3},
     year = {2020},
     doi = {10.5802/aif.3329},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3329/}
}
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/articles/10.5802/aif.3329/

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