An estimate for the average number of common zeros of Laplacian eigenfunctions
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Dmitri Akhiezer and Boris Kazarnovskii
Translated by: the authors - Trans. Moscow Math. Soc. 2017, 123-130
- DOI: https://doi.org/10.1090/mosc/269
- Published electronically: December 1, 2017
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Abstract:
On a compact Riemannian manifold $M$ of dimension $n$, we consider $n$ eigenfunctions of the Laplace operator $\Delta$ with eigenvalue $\lambda$. If $M$ is homogeneous under a compact Lie group preserving the metric then we prove that the average number of common zeros of $n$ eigenfunctions does not exceed $c(n)\lambda ^{n/2}\textrm {vol} M$, the expression known from the celebrated Weyl’s law. Moreover, if the isotropy representation is irreducible, then the estimate turns into the equality. The constant $c(n)$ is explicitly given. The method of proof is based on the application of Crofton’s formula for the sphere.References
- Dmitri Akhiezer and Boris Kazarnovskii, On common zeros of eigenfunctions of the Laplace operator, Abh. Math. Semin. Univ. Hambg. 87 (2017), no. 1, 105–111. MR 3623828, DOI 10.1007/s12188-016-0138-1
- J. C. Álvarez Paiva and E. Fernandes, Gelfand transforms and Crofton formulas, Selecta Math. (N.S.) 13 (2007), no. 3, 369–390. MR 2383600, DOI 10.1007/s00029-007-0045-5
- Vladimir I. Arnold, Arnold’s problems, Springer-Verlag, Berlin; PHASIS, Moscow, 2004. Translated and revised edition of the 2000 Russian original; With a preface by V. Philippov, A. Yakivchik and M. Peters. MR 2078115
- Shiu Yuen Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), no. 1, 43–55. MR 397805, DOI 10.1007/BF02568142
- R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
- 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
- Israel M. Gelfand and Mikhail M. Smirnov, Lagrangians satisfying Crofton formulas, Radon transforms, and nonlocal differentials, Adv. Math. 109 (1994), no. 2, 188–227. MR 1304752, DOI 10.1006/aima.1994.1086
- V. M. Gichev, Metric properties in the mean of polynomials on compact isotropy irreducible homogeneous spaces, Anal. Math. Phys. 3 (2013), no. 2, 119–144. MR 3057234, DOI 10.1007/s13324-012-0051-4
- Ralph Howard, The kinematic formula in Riemannian homogeneous spaces, Mem. Amer. Math. Soc. 106 (1993), no. 509, vi+69. MR 1169230, DOI 10.1090/memo/0509
- Victor Ivrii, 100 years of Weyl’s law, Bull. Math. Sci. 6 (2016), no. 3, 379–452. MR 3556544, DOI 10.1007/s13373-016-0089-y
- O. V. Manturov, Homogeneous Riemannian spaces with an irreducible rotation group, Trudy Sem. Vektor. Tenzor. Anal. 13 (1966), 68–145 (Russian). MR 0210031
- Luis A. Santaló, Integral geometry and geometric probability, Encyclopedia of Mathematics and its Applications, Vol. 1, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac. MR 0433364
- Tsunero Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380–385. MR 198393, DOI 10.2969/jmsj/01840380
- Joseph A. Wolf, The goemetry and structure of isotropy irreducible homogeneous spaces, Acta Math. 120 (1968), 59–148. MR 223501, DOI 10.1007/BF02394607
Bibliographic Information
- Dmitri Akhiezer
- Affiliation: Institute for Information Transmission Problems 19 B. Karetny per., 127994, Moscow, Russia
- Email: akhiezer@iitp.ru
- Boris Kazarnovskii
- Affiliation: Institute for Information Transmission Problems 19 B. Karetny per., 127994, Moscow, Russia
- Email: kazbori@gmail.com
- Published electronically: December 1, 2017
- Additional Notes: The research was carried out at the Institute for Information Transmission Problems under support by the Russian Foundation of Sciences, grant No. 14-50-00150
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Moscow Math. Soc. 2017, 123-130
- MSC (2010): Primary 53C30, 58J05
- DOI: https://doi.org/10.1090/mosc/269
- MathSciNet review: 3738081
Dedicated: To Ernest Borisovich Vinberg on the occasion of his 80th birthday