Variation of Laplace spectra of compact “nearly” hyperbolic surfaces

Mukherjee, Mayukh

doi:10.1016/j.crma.2016.11.019

Differential geometry

Variation of Laplace spectra of compact “nearly” hyperbolic surfaces
[Variation du spectre de Laplace des surfaces compactes « presque » hyperboliques]

Présenté par : the Editorial Board

Mukherjee, Mayukh ¹

¹ Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany

Comptes Rendus. Mathématique, Tome 355 (2017) no. 2, pp. 216-221.

Résumé
Abstract

Nous utilisons l'analyticité réelle du flot de Ricci par rapport au temps, démontrée par B. Kotschwar, pour étendre un résultat de P. Buser. Précisément, nous montrons que le spectre de Laplace des surfaces compactes, orientables, de courbure négative, de même genre $γ \geq 2$ , même aire et mêmes bornes pour la courbure, varie de « façon contrôlée ». Nous donnons une estimation quantitative de cette variation dans notre théorème principal. Notre outil technique de base est une formule variationnelle donnant la dérivée d'une branche de valeur propre sous l'action du flot de Ricci normalisé. Par analogie, nous indiquons comment le résultat d'analyticité réelle ci-dessus peut conduire à des conclusions inattendues sur les propriétés du spectre des métriques génériques sur une surface compacte, de genre $γ \geq 2$ .

We use the real analyticity of the Ricci flow with respect to time proved by B. Kotschwar to extend a result of P. Buser, namely, we prove that the Laplace spectra of negatively curved compact orientable surfaces having the same genus $γ \geq 2$ , the same area and the same curvature bounds vary in a “controlled way”, of which we give a quantitative estimate in our main theorem. The basic technical tool is a variational formula that provides the derivative of an eigenvalue branch under the normalized Ricci flow. In a related manner, we also observe how the above-mentioned real analyticity result can lead to unexpected conclusions concerning the spectral properties of generic metrics on a compact surface of genus $γ \geq 2$ .

Reçu le : 2015-12-31
Accepté le : 2016-12-09
Publié le : 2016-12-27

DOI : 10.1016/j.crma.2016.11.019

Affiliations des auteurs :

Mukherjee, Mayukh ¹

¹ Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany

@article{CRMATH_2017__355_2_216_0,
     author = {Mukherjee, Mayukh},
     title = {Variation of {Laplace} spectra of compact {\textquotedblleft}nearly{\textquotedblright} hyperbolic surfaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {216--221},
     publisher = {Elsevier},
     volume = {355},
     number = {2},
     year = {2017},
     doi = {10.1016/j.crma.2016.11.019},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2016.11.019/}
}

TY  - JOUR
AU  - Mukherjee, Mayukh
TI  - Variation of Laplace spectra of compact “nearly” hyperbolic surfaces
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 216
EP  - 221
VL  - 355
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2016.11.019/
DO  - 10.1016/j.crma.2016.11.019
LA  - en
ID  - CRMATH_2017__355_2_216_0
ER  -

%0 Journal Article
%A Mukherjee, Mayukh
%T Variation of Laplace spectra of compact “nearly” hyperbolic surfaces
%J Comptes Rendus. Mathématique
%D 2017
%P 216-221
%V 355
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2016.11.019/
%R 10.1016/j.crma.2016.11.019
%G en
%F CRMATH_2017__355_2_216_0

Mukherjee, Mayukh. Variation of Laplace spectra of compact “nearly” hyperbolic surfaces. Comptes Rendus. Mathématique, Tome 355 (2017) no. 2, pp. 216-221. doi : 10.1016/j.crma.2016.11.019. http://www.numdam.org/articles/10.1016/j.crma.2016.11.019/

Bibliographie
Cité par

[1] Bando, S. Real analyticity of solutions of Hamilton's equation, Math. Z., Volume 195 (1987) no. 1, pp. 93-97

[2] Buser, P. Geometry and Spectra of Compact Riemann Surfaces, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, USA, 2010 (Reprint of the 1992 edition)

[3] Cao, X. Eigenvalues of $(- Δ + \frac{R}{2})$ on manifolds with nonnegative curvature operator, J. Geom. Anal., Volume 17 (2007) no. 3, pp. 425-433

[4] Cao, X. First eigenvalues of geometric operators under the Ricci flow, Proc. Amer. Math. Soc., Volume 136 (2008) no. 11, pp. 4075-4078

[5] Chow, B.; Knopf, D. The Ricci Flow: An Introduction, Mathematical Surveys and Monographs, vol. 110, American Mathematical Society, Providence, RI, USA, 2004

[6] DeTurck, D. Deforming metrics in the direction of their Ricci tensors, J. Differ. Geom., Volume 18 (1983) no. 1, pp. 157-162

[7] Di Cerbo, L.F. Eigenvalues of the Laplacian under the Ricci flow, Rend. Mat. Appl. (7), Volume 27 (2007) no. 2, pp. 183-195

[8] Hamilton, R. Three-manifolds with positive Ricci curvature, J. Differ. Geom., Volume 17 (1982) no. 2, pp. 255-306

[9] Kato, T. Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995 (Reprint of the 1980 edition)

[10] Kotschwar, B. Backwards uniqueness for the Ricci flow, Int. Math. Res. Not., Volume 2010 (2010) no. 21, pp. 4064-4097

[11] Kotschwar, B. Time-analyticity of solutions to the Ricci flow, Amer. J. Math., Volume 137 (2015) no. 2, pp. 535-576

[12] Kriegl, A.; Michor, P.; Rainer, A. Denjoy–Carleman differentiable perturbation of polynomials and unbounded operators, Integral Equ. Oper. Theory, Volume 71 (2011) no. 3, pp. 407-416

[13] Uhlenbeck, K. Generic properties of eigenfunctions, Amer. J. Math., Volume 98 (1976) no. 4, pp. 1059-1078

[14] Wolpert, S. Spectral limits for hyperbolic surfaces, II, Invent. Math., Volume 108 (1992) no. 1, pp. 91-129

[15] Wolpert, S. Disappearance of cusp forms in special families, Ann. Math. (2), Volume 139 (1994) no. 2, pp. 239-291

Cité par Sources :