Abstract
We give a new definition of a Laplace operator for Finsler metric as an average with regard to an angle measure of the second directional derivatives. This definition uses a dynamical approach due to Foulon that does not require the use of connections or local coordinates. We give explicit representations and computations of spectral data for this operator in the case of Katok-Ziller metrics on the sphere and the torus.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications Inc., New York, 1992, Reprint of the 1972 edition.
H. Akbar-Zadeh, Sur les espaces de Finsler à courbures sectionnelles constantes, Académie Royale de Belgique. Bulletin de la Classe des Sciences 74 (1988), 281–322.
M. Anastasiei and H. Kawaguchi, Absolute energy of a Finsler space, The Tensor Society. Tensor 53 (1993), Commemoration Volume I, International Conference on Differential Geometry and its Applications (Bucharest, 1992), pp. 108–113.
M. T. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Annals of Mathematics 121 (1985), 429–461.
P. L. Antonelli and T. J. Zastawniak, Stochastic calculus on Finsler manifolds and an application in biology, Nonlinear World 1 (1994), 149–171.
V. Bangert and Y. Long, The existence of two closed geodesics on every Finsler 2-sphere, Mathematische Annalen 346 (2010), 335–366.
D. Bao, S.-S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics, Vol. 200, Springer-Verlag, New York, 2000.
D. Bao and B. Lackey, A Hodge decomposition theorem for Finsler spaces, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 323 (1996), 51–56.
D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds, Journal of Differential Geometry 66 (2004), 377–435.
T. Barthelmé, A new Laplace operator in Finsler geometry and periodic orbits of Anosov flows, Ph.D. thesis, Université de Strasbourg, 2012.
P. H. Bérard, Spectral geometry: Direct and Inverse Problems, with appendixes by Gérard Besson, and by Bérard and Marcel Berger, Lecture Notes in Mathematics, Vol. 1207, Springer-Verlag, Berlin, 1986.
M. Berger, P. Gauduchon and E. Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin, 1971.
R. L. Bryant, Finsler structures on the 2-sphere satisfying K = 1, in Finsler Geometry (Seattle, WA, 1995), Contemporary Mathematics, Vol. 196, American Mathematical Society, Providence, RI, 1996, pp. 27–41.
R. L. Bryant, Projectively flat Finsler 2-spheres of constant curvature, Selecta Mathematica 3 (1997), 161–203.
R. L. Bryant, Geodesically reversible Finsler 2-spheres of constant curvature, in Inspired by S. S. Chern, Nankai Tracts in Mathematics, Vol. 11, World Scientific Publishing, Hackensack, NJ, 2006, pp. 95–111.
D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, Vol. 33, American Mathematical Society, Providence, RI, 2001.
P. Centore, A mean-value Laplacian for Finsler spaces, in The Theory of Finslerian Laplacians and Applications, Mathematics and its Applications, Vol. 459, Kluwer Academic Publishing, Dordrecht, 1998, pp. 151–186.
P. Centore, Finsler Laplacians and minimal-energy maps, International Journal of Mathematics 11 (2000), 1–13.
I. Chavel, Eigenvalues in Riemannian Geometry, Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk, Pure and Applied Mathematics, Vol. 115, Academic Press Inc., Orlando, FL, 1984.
I. Chavel and E. A. Feldman, Isoperimetric constants and large time heat diffusion in Riemannian manifolds, in Differential Geometry: Riemannian Geometry (Los Angeles, CA, 1990), Proceedings of Symposia in Pure Mathematics, Vol. 54, American Mathematical Society, Providence, RI, 1993, pp. 111–121.
M. Crampon, Entropies of strictly convex projective manifolds, Journal of Modern Dynamics 3 (2009), 511–547.
E. B. Davies, Heat kernel bounds, conservation of probability and the Feller property, Journal d’Analayse Mathématique 58 (1992), 99–119, Festschrift on the occasion of the 70th birthday of Shmuel Agmon.
D. Egloff, Some new developments in Finsler geometry, Ph.D. thesis, Univ. Fribourg, 1995.
D. Egloff, On the dynamics of uniform Finsler manifolds of negative flag curvature, Annals of Global Analysis and Geometry 15 (1997), 101–116.
D. Egloff, Uniform Finsler Hadamard manifolds, Annales de l’Institut Henri Poincaré. Physique Théorique 66 (1997), 323–357.
P. Foulon, Personal communication.
P. Foulon, Géométrie des équations différentielles du second ordre, Annales de l’Institut Henri Poincaré. Physique Théorique 45 (1986), 1–28.
P. Foulon, Estimation de l’entropie des systèmes lagrangiens sans points conjugués, Annales de l’Institut Henri Poincaré. Physique Théorique 57 (1992), 117–146, With an appendix, “About Finsler geometry”, in English.
P. Foulon, Locally symmetric Finsler spaces in negative curvature, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 324 (1997), 1127–1132.
S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, third edn., Universitext, Springer-Verlag, Berlin, 2004.
A. Grigor’yan, Heat kernels on weighted manifolds and applications, in The Ubiquitous Heat Kernel, Contemporary Mathematics, Vol. 398, American Mathematical Society, Providence, RI, 2006, pp. 93–191.
J. Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences. Séries A et B 270 (1970), A1645–A1648.
R. D. Holmes and A. C. Thompson, n-dimensional area and content in Minkowski spaces, Pacific Journal of Mathematics 85 (1979), 77–110.
T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.
A. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems, Izvestiya Akademii Nauk SSSR. Seriya au]Matematicheskaya 37 (1973), 539–576.
F. Ledrappier, Ergodic properties of Brownian motion on covers of compact negatively-curve manifolds, Boletim da Sociedade Brasileira Matemática 19 (1988), 115–140.
R. Narasimhan, Analysis on Real and Complex Manifolds, Advanced Studies in Pure Mathematics, Vol. 1, Masson & Cie, Éditeurs, Paris, 1968.
H.-B. Rademacher, A sphere theorem for non-reversible Finsler metrics, Mathematische Annalen 328 (2004), 373–387.
M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975.
M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978.
M. Reed and B. Simon, Methods of ModernMathematical Physics. I. Functional Analysis, second edn., Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980.
Z. Shen, The non-linear Laplacian for Finsler manifolds, in The Theory of Finslerian Laplacians and Applications, Mathematics and its Applications, Vol. 459, Kluwer Academic Publishing, Dordrecht, 1998, pp. 187–198.
D. Sullivan, The Dirichlet problem at infinity for a negatively curved manifold, Journal of Differential Geometry 18 (1983), 723–732 (1984).
W. Ziller, Geometry of the Katok examples, Ergodic Theory and Dynamical Systems 3 (1983), 135–157.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barthelmé, T. A natural Finsler-Laplace operator. Isr. J. Math. 196, 375–412 (2013). https://doi.org/10.1007/s11856-012-0168-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-012-0168-z