[Limite du nombre de domaines nodaux des fonctions propres de variétés Kaluza–Klein génériques en dimension 3]
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.
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Keywords: Eigenfunction of the Laplacian, Principal bundle, Kaluza–Klein metric, Nodal domain
Mot clés : fonction propre du Laplacien, fibré principal, métrique de Kaluza–Klein, domaine nodal
Jung, Junehyuk 1 ; Zelditch, Steve 2
CC-BY-ND 4.0
@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/}
}
TY - JOUR AU - Jung, Junehyuk AU - Zelditch, Steve TI - Boundedness of the number of nodal domains for eigenfunctions of generic Kaluza–Klein 3-folds JO - Annales de l'Institut Fourier PY - 2020 SP - 971 EP - 1027 VL - 70 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3329/ DO - 10.5802/aif.3329 LA - en ID - AIF_2020__70_3_971_0 ER -
%0 Journal Article %A Jung, Junehyuk %A Zelditch, Steve %T Boundedness of the number of nodal domains for eigenfunctions of generic Kaluza–Klein 3-folds %J Annales de l'Institut Fourier %D 2020 %P 971-1027 %V 70 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3329/ %R 10.5802/aif.3329 %G en %F 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/articles/10.5802/aif.3329/
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