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/

[1] Arias, F. A.; Malakhaltsev, Mikhail 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] Barnett, Alex; Konrad, Kyle; Jin, Matthew Experimental Nazarov-Sodin constants, genus, and percolation on nodal domains for 2D and 3D random waves (2017) (in preparation)

[3] Bérard-Bergery, Lionel; Bourguignon, Jean-Pierre Laplacians and Riemannian submersions with totally geodesic fibres, Ill. J. Math., Volume 26 (1982) no. 2, pp. 181-200 | MR 650387

[4] Courant, Richard; Hilbert, David Methods of mathematical physics. Vol. I, Interscience Publishers, 1953, xv+561 pages | MR 0065391

[5] Demailly, Jean-Pierre 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] Enciso, Alberto; Peralta-Salas, Daniel 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] Fukui, Toshizumi; Nuño-Ballesteros, Juan J. Isolated singularities of binary differential equations of degree n, Publ. Mat., Barc., Volume 56 (2012) no. 1, pp. 65-89 | Article | MR 2918184 | Zbl 1282.37017

[8] Ghosh, Amit; Reznikov, Andre; Sarnak, Peter Nodal domains of Maass forms I, Geom. Funct. Anal., Volume 23 (2013) no. 5, pp. 1515-1568 | Article | MR 3102912

[9] Ghosh, Amit; Reznikov, Andre; Sarnak, Peter Nodal domains of Maass forms, II, Am. J. Math., Volume 139 (2017) no. 5, pp. 1395-1447 | Article | MR 3702502

[10] Iwaniec, Henryk; Kowalski, Emmanuel Analytic number theory, Colloquium Publications, Volume 53, American Mathematical Society, 2004, xii+615 pages | Article | MR 2061214

[11] Jang, Seung uk; Jung, Junehyuk 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] Jung, Junehyuk; Young, Matthew P. Sign changes of the Eisenstein series on the critical line, Int. Math. Res. Not. (2019) no. 3, pp. 641-672 | Article | MR 3910468

[13] Jung, Junehyuk; Zelditch, Steve 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] Jung, Junehyuk; Zelditch, Steve 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] Jung, Junehyuk; Zelditch, Steve Topology of the nodal set of random equivariant spherical harmonics on 𝕊 3 (2019) (https://arxiv.org/abs/1908.00979, to appear in Int. Math. Res. Not.)

[16] Lewy, Hans 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] Magee, Michael Arithmetic, zeros, and nodal domains on the sphere, Commun. Math. Phys., Volume 338 (2015) no. 3, pp. 919-951 | Article | MR 3355806

[18] Stern, Antonie Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen. Math.- naturwiss. Diss., Göttingen, 30 S (1925)., 1925 | Zbl 51.0356.01

[19] Strebel, Kurt Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Volume 5, Springer, 1984, xii+184 pages | Article | MR 743423

[20] Uhlenbeck, Karen Generic properties of eigenfunctions, Am. J. Math., Volume 98 (1976) no. 4, pp. 1059-1078 | Article | MR 0464332 | Zbl 0355.58017

[21] Vilms, Jaak Totally geodesic maps, J. Differ. Geom., Volume 4 (1970), pp. 73-79 | MR 0262984

[22] Zelditch, Steve Logarithmic lower bound on the number of nodal domains, J. Spectr. Theory, Volume 6 (2016) no. 4, pp. 1047-1086 | Article | MR 3584194