no. 3
On deformations of the spectrum of a Finsler-Laplacian that preserve the length spectrum
[Sur des déformations du spectre d’un opérateur de Finsler-Laplace préservant le spectre des longueurs]
Barthelmé, Thomas1, 2
1 Pennsylvania State University, State College, PA, USA
2 Queen’s University, Kingston, ON, Canada
Bulletin de la Société Mathématique de France, Tome 145 (2017) no. 3, pp. 421-448

The main result of this article is the construction of non-reversible Finsler metrics in negative curvature such that 4λ1>h2, where λ1 is the bottom of the L2-spectrum of a previously defined Finsler-Laplacian and h the topological entropy of the flow. This gives a counter-example to a classical inequality in Riemannian geometry. We also show that the spectrum of that Finsler-Laplacian can detect changes in the Finsler metric that the marked length spectrum cannot.

Le résultat principal de cet article est la construction d’une famille de métriques de Finsler, non-réversible, en courbure négative satisfaisant 4λ1>h2, où λ1 est le bas du spectre L2 d’un laplacien en géométrie de Finsler et h est l’entropie topologique du flot géodésique. Ce résultat fournit un contre-exemple, pour les métriques de Finsler, à une inégalité classique de géométrie riemannienne. Nous montrons également que le spectre de ce laplacien détecte certains changements de la métrique qui sont invisible pour le spectre des longueurs.

Reçu le :
Accepté le :
Publié le :
DOI : 10.24033/bsmf.2743
Classification : 58J60, 53C60
Keywords: Finsler-Laplacian, topological entropy, length spectrum
Mots-clés : Laplacien Finsler, entropy topologique, spectre des longueurs

Barthelmé, Thomas 1, 2

1 Pennsylvania State University, State College, PA, USA
2 Queen’s University, Kingston, ON, Canada 
@article{BSMF_2017__145_3_421_0,
     author = {Barthelm\'e, Thomas},
     title = {On deformations of the spectrum of a {Finsler-Laplacian} that preserve the length spectrum},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {421--448},
     year = {2017},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {145},
     number = {3},
     doi = {10.24033/bsmf.2743},
     mrnumber = {3766116},
     zbl = {1411.53058},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2743/}
}
TY  - JOUR
AU  - Barthelmé, Thomas
TI  - On deformations of the spectrum of a Finsler-Laplacian that preserve the length spectrum
JO  - Bulletin de la Société Mathématique de France
PY  - 2017
SP  - 421
EP  - 448
VL  - 145
IS  - 3
PB  - Société mathématique de France
UR  - https://www.numdam.org/articles/10.24033/bsmf.2743/
DO  - 10.24033/bsmf.2743
LA  - en
ID  - BSMF_2017__145_3_421_0
ER  - 
%0 Journal Article
%A Barthelmé, Thomas
%T On deformations of the spectrum of a Finsler-Laplacian that preserve the length spectrum
%J Bulletin de la Société Mathématique de France
%D 2017
%P 421-448
%V 145
%N 3
%I Société mathématique de France
%U https://www.numdam.org/articles/10.24033/bsmf.2743/
%R 10.24033/bsmf.2743
%G en
%F BSMF_2017__145_3_421_0
Barthelmé, Thomas. On deformations of the spectrum of a Finsler-Laplacian that preserve the length spectrum. Bulletin de la Société Mathématique de France, Tome 145 (2017) no. 3, pp. 421-448. doi: 10.24033/bsmf.2743

Barthelmé, Thomas A new Laplace operator in Finsler geometry and periodic orbits of Anosov flows, Ph. D. Thesis , Université de Strasbourg (2012)

Barthelmé, Thomas A natural Finsler-Laplace operator, Israel J. Math., Volume 196 (2013), pp. 375-412 | MR | Zbl | DOI

Barthelmé, Thomas; Colbois, Bruno Eigenvalue control for a Finsler-Laplace operator, Ann. Global Anal. Geom., Volume 44 (2013), pp. 43-72 | MR | Zbl | DOI

Barthelmé, Thomas; Colbois, Bruno; Crampon, Mickaël; Verovic, Patrick Laplacian and spectral gap in regular Hilbert geometries, Tohoku Math. J., Volume 66 (2014), pp. 377-407 | MR | Zbl | DOI

Besson, G.; Courtois, G.; Gallot, S. Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal., Volume 5 (1995), pp. 731-799 | MR | Zbl | DOI

Bonnesen, T.; Fenchel, W. Theory of convex bodies, BCS Associates, Moscow, ID, 1987, 172 pages | MR | Zbl

Colin de Verdière, Yves Spectre du laplacien et longueurs des géodésiques périodiques. I, II, Compos. math., Volume 27 (1973), p. 27 | MR | Zbl | Numdam

Croke, Christopher B. Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv., Volume 65 (1990), pp. 150-169 | MR | Zbl | DOI

Donnelly, Harold; Fefferman, Charles Nodal sets of eigenfunctions on Riemannian manifolds, Invent. math., Volume 93 (1988), pp. 161-183 | MR | Zbl | DOI

Eberlein, Patrick Geometry of 2-step nilpotent groups with a left invariant metric, Ann. Sci. École Norm. Sup., Volume 27 (1994), pp. 611-660 | Numdam | MR | Zbl | DOI

Egloff, Daniel On the dynamics of uniform Finsler manifolds of negative flag curvature, Ann. Global Anal. Geom., Volume 15 (1997), pp. 101-116 | MR | Zbl | DOI

Fang, Yong Thermodynamic invariants of Anosov flows and rigidity, Discrete Contin. Dyn. Syst., Volume 24 (2009), pp. 1185-1204 | MR | Zbl | DOI

Fang, Yong; Foulon, Patrick On Finsler manifolds of negative flag curvature, J. Topol. Anal., Volume 7 (2015), pp. 483-504 | MR | Zbl | DOI

Foulon, Patrick Géométrie des équations différentielles du second ordre, Ann. Inst. H. Poincaré Phys. Théor., Volume 45 (1986), pp. 1-28 | Numdam | MR | Zbl

Gordon, Carolyn S. When you can’t hear the shape of a manifold, Math. Intelligencer, Volume 11 (1989), pp. 39-47 | MR | Zbl | DOI

Gornet, Ruth The marked length spectrum vs. the Laplace spectrum on forms on Riemannian nilmanifolds, Comment. Math. Helv., Volume 71 (1996), pp. 297-329 | MR | Zbl | DOI

Guillemin, Victor The Radon transform on Zoll surfaces, Advances in Math., Volume 22 (1976), pp. 85-119 | MR | Zbl | DOI

He, Qun; Zeng, Fanqi; Zeng, Daxiao On the first eigenvalue of the mean finsler-laplacian, Acta Mathematica Scientia, Volume 37 (2017), pp. 1162-1172 http://www.sciencedirect.com/... | MR | Zbl | DOI

Katok, Anatole Ergodic perturbations of degenerate integrable Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat., Volume 37 (1973), pp. 539-576 | MR | Zbl

Katok, Anatole Four applications of conformal equivalence to geometry and dynamics, Ergodic Theory Dynam. Systems, Volume 8* (1988), pp. 139-152 | MR | Zbl | DOI

Katok, Anatole; Hasselblatt, Boris Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge Univ. Press, Cambridge, 1995, 802 pages | MR | Zbl | DOI

Korevaar, Nicholas Upper bounds for eigenvalues of conformal metrics, J. Differential Geom., Volume 37 (1993), pp. 73-93 http://projecteuclid.org/euclid.jdg/1214453423 | MR | Zbl

Ledrappier, F. Harmonic measures and Bowen-Margulis measures, Israel J. Math., Volume 71 (1990), pp. 275-287 | MR | Zbl | DOI

Lin, Fang-Hua Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., Volume 44 (1991), pp. 287-308 | MR | Zbl | DOI

Manning, Anthony Topological entropy for geodesic flows, Ann. of Math., Volume 110 (1979), pp. 567-573 | MR | Zbl | DOI

Otal, Jean-Pierre Le spectre marqué des longueurs des surfaces à courbure négative, Ann. of Math., Volume 131 (1990), pp. 151-162 | MR | Zbl | DOI

Handbook of Hilbert geometry (Papadopoulos, Athanase; Troyanov, Marc, eds.), IRMA Lectures in Mathematics and Theoretical Physics, 22, European Mathematical Society (EMS), Zürich, 2014, 452 pages | MR | Zbl | DOI

Schoen, R.; Yau, S.-T. Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994, 235 pages | MR | Zbl

Thompson, A. C. Minkowski geometry, Encyclopedia of Mathematics and its Applications, 63, Cambridge Univ. Press, Cambridge, 1996, 346 pages | MR | Zbl | DOI

Ziller, Wolfgang Geometry of the Katok examples, Ergodic Theory Dynam. Systems, Volume 3 (1983), pp. 135-157 | MR | Zbl | DOI

Cité par Sources :