Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders
HTML articles powered by AMS MathViewer
- by Pierre Bérard, Bernard Helffer and Rola Kiwan PDF
- Proc. Amer. Math. Soc. 150 (2022), 439-453 Request permission
Abstract:
The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, Möbius strips, and so forth. A natural toy model for further investigations is the flat Klein bottle, a non-orientable surface with Euler characteristic $0$, and particularly the Klein bottle associated with the square torus, whose eigenvalues have higher multiplicities. In this note, we prove that the only Courant-sharp eigenvalues of the flat Klein bottle associated with the square torus (resp. with square fundamental domain) are the first and second eigenvalues. We also consider the flat cylinders $(0,\pi ) \times \mathbb {S}^1_r$ where $r \in \left \lbrace 0.5,1 \right \rbrace$ is the radius of the circle $\mathbb {S}^1_r$, and we show that the only Courant-sharp Dirichlet eigenvalues of these cylinders are the first and second eigenvalues.References
- Ram Band, Michael Bersudsky, and David Fajman, Courant-sharp eigenvalues of Neumann 2-rep-tiles, Lett. Math. Phys. 107 (2017), no. 5, 821–859. MR 3633026, DOI 10.1007/s11005-016-0926-7
- Pierre Bérard and Bernard Helffer. Nodal sets of eigenfunctions, Antonie Stern’s results revisited. In Séminaire de théorie spectrale et géométrie, volume 32, pages 1–37. Université de Grenoble, 2014–2015.
- Pierre Bérard and Bernard Helffer, Dirichlet eigenfunctions of the square membrane: Courant’s property, and A. Stern’s and Å. Pleijel’s analyses, Analysis and geometry, Springer Proc. Math. Stat., vol. 127, Springer, Cham, 2015, pp. 69–114. MR 3445517, DOI 10.1007/978-3-319-17443-3_{6}
- Pierre Bérard and Bernard Helffer, Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle, Lett. Math. Phys. 106 (2016), no. 12, 1729–1789. MR 3569644, DOI 10.1007/s11005-016-0819-9
- Pierre Bérard, Bernard Helffer, and Rola Kiwan, Courant-sharp property for Dirichlet eigenfunctions on the Möbius strip, Port. Math. 78 (2021), no. 1, 1–41. MR 4269391, DOI 10.4171/pm/2059
- Pierre Bérard and Daniel Meyer, Inégalités isopérimétriques et applications, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 3, 513–541 (French). MR 690651
- Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313
- Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine, Riemannian geometry, 3rd ed., Universitext, Springer-Verlag, Berlin, 2004. MR 2088027, DOI 10.1007/978-3-642-18855-8
- K. Gittins and B. Helffer, Courant-sharp Robin eigenvalues for the square: the case with small Robin parameter, Ann. Math. Qué. 44 (2020), no. 1, 91–123 (English, with English and French summaries). MR 4071872, DOI 10.1007/s40316-019-00120-7
- Katie Gittins and Bernard Helffer, Courant-sharp Robin eigenvalues for the square and other planar domains, Port. Math. 76 (2019), no. 1, 57–100. MR 4016623, DOI 10.4171/PM/2027
- B. Helffer, T. Hoffmann-Ostenhof, and S. Terracini, Nodal domains and spectral minimal partitions, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 1, 101–138. MR 2483815, DOI 10.1016/j.anihpc.2007.07.004
- Bernard Helffer and Thomas Hoffmann-Ostenhof, Spectral minimal partitions for a thin strip on a cylinder or a thin annulus like domain with Neumann condition, Operator methods in mathematical physics, Oper. Theory Adv. Appl., vol. 227, Birkhäuser/Springer Basel AG, Basel, 2013, pp. 107–115. MR 3050161, DOI 10.1007/978-3-0348-0531-5_{5}
- Bernard Helffer and Thomas Hoffmann-Ostenhof, Minimal partitions for anisotropic tori, J. Spectr. Theory 4 (2014), no. 2, 221–233. MR 3232810, DOI 10.4171/JST/68
- Bernard Helffer and Rola Kiwan, Dirichlet eigenfunctions in the cube, sharpening the Courant nodal inequality, Functional analysis and operator theory for quantum physics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2017, pp. 353–371. MR 3677019
- Bernard Helffer and Mikael Persson Sundqvist, On nodal domains in Euclidean balls, Proc. Amer. Math. Soc. 144 (2016), no. 11, 4777–4791. MR 3544529, DOI 10.1090/proc/13098
- Bernard Helffer and Mikael Persson Sundqvist, Nodal domains in the square—the Neumann case, Mosc. Math. J. 15 (2015), no. 3, 455–495, 605 (English, with English and Russian summaries). MR 3427435, DOI 10.17323/1609-4514-2015-15-3-455-495
- Hugh Howards, Michael Hutchings, and Frank Morgan, The isoperimetric problem on surfaces, Amer. Math. Monthly 106 (1999), no. 5, 430–439. MR 1699261, DOI 10.2307/2589147
- Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR 0152974
- Corentin Léna, Courant-sharp eigenvalues of a two-dimensional torus, C. R. Math. Acad. Sci. Paris 353 (2015), no. 6, 535–539 (English, with English and French summaries). MR 3348988, DOI 10.1016/j.crma.2015.03.014
- Corentin Léna, Courant-sharp eigenvalues of the three-dimensional square torus, Proc. Amer. Math. Soc. 144 (2016), no. 9, 3949–3958. MR 3513551, DOI 10.1090/proc/13148
- Corentin Léna, Pleijel’s nodal domain theorem for Neumann and Robin eigenfunctions, Ann. Inst. Fourier (Grenoble) 69 (2019), no. 1, 283–301 (English, with English and French summaries). MR 3973450
- Josef Leydold, On the number of nodal domains of spherical harmonics, Topology 35 (1996), no. 2, 301–321. MR 1380499, DOI 10.1016/0040-9383(95)00028-3
- Jaak Peetre, A generalization of Courant’s nodal domain theorem, Math. Scand. 5 (1957), 15–20. MR 92917, DOI 10.7146/math.scand.a-10484
- Åke Pleijel, Remarks on Courant’s nodal line theorem, Comm. Pure Appl. Math. 9 (1956), 543–550. MR 80861, DOI 10.1002/cpa.3160090324
- Iosif Polterovich, Pleijel’s nodal domain theorem for free membranes, Proc. Amer. Math. Soc. 137 (2009), no. 3, 1021–1024. MR 2457442, DOI 10.1090/S0002-9939-08-09596-8
- Joseph A. Wolf, Spaces of constant curvature, 6th ed., AMS Chelsea Publishing, Providence, RI, 2011. MR 2742530, DOI 10.1090/chel/372
Additional Information
- Pierre Bérard
- Affiliation: Université Grenoble Alpes and CNRS, Institut Fourier, CS 40700, F38058 Grenoble Cedex 9, France
- MR Author ID: 34955
- ORCID: 0000-0001-8712-9269
- Email: pierrehberard@gmail.com
- Bernard Helffer
- Affiliation: Laboratoire Jean Leray, Université de Nantes and CNRS, F44322 Nantes Cedex, France; and LMO (Université Paris-Sud)
- MR Author ID: 83860
- Email: Bernard.Helffer@univ-nantes.fr
- Rola Kiwan
- Affiliation: American University in Dubai, P.O. Box 28282, Dubai, United Arab Emirates
- MR Author ID: 801127
- ORCID: 0000-0003-2956-9922
- Email: rkiwan@aud.edu
- Received by editor(s): November 12, 2020
- Received by editor(s) in revised form: April 8, 2021
- Published electronically: October 25, 2021
- Communicated by: Tanya Christiansen
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 439-453
- MSC (2020): Primary 58C40, 49Q10
- DOI: https://doi.org/10.1090/proc/15620
- MathSciNet review: 4335889