Approximate rigidity of the marked length spectrum

Butt, Karen

Mathematics Research Reports, Tome 4 (2023), pp. 63-82

We report on recent work investigating the extent to which finitely many closed geodesics approximately determine a negatively curved metric on a closed manifold. It is known in certain cases—and conjectured to be true in general—that the lengths of all closed geodesics (as a function of their free homotopy classes) determine the underlying negatively curved metric up to isometry. This length function is known as the marked length spectrum. Here, we consider certain pairs of Riemannian manifolds whose marked length spectra agree—only approximately—on a finite set of closed geodesics. We report on our recent results which show the two metrics are “almost isometric". More precisely, we show the metrics are bi-Lipschitz equivalent with constant close to 1, and we obtain estimates for these constants depending only on concrete Riemannian data.

Classification : 37D40, 37D20, 37C27, 53C24, 53C22
Keywords: marked length spectrum, closed geodesics, geodesic flow, negative curvature, locally symmetric spaces, rigidity

[1] Aravinda, C. S.; Farrell, F. T. Exotic negatively curved structures on Cayley hyperbolic manifolds, J. Differ. Geom., Volume 63 (2003) no. 1, pp. 41-62 | DOI | MR | Zbl

[2] Aravinda, C. S.; Farrell, F. T. Exotic structures and quaternionic hyperbolic manifolds, Algebraic groups and arithmetic (Tata Institute of Fundamental Research Studies in Mathematics), Volume 17, Tata Institute of Fundamental Research, 2004, pp. 507-524 | Zbl

[3] Ballmann, Werner Lectures on spaces of nonpositive curvature, 25, Springer, 1995 | DOI | Numdam

[4] Benedetti, Riccardo; Petronio, Carlo Lectures on hyperbolic geometry, Springer, 1992 | DOI

[5] Bessières, Laurent; Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre Differentiable rigidity under Ricci curvature lower bound, Duke Math. J., Volume 161 (2012) no. 1, pp. 29-67 | MR | Zbl

[6] Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal., Volume 5 (1995) no. 5, pp. 731-799 | DOI

[7] Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre Minimal entropy and Mostow’s rigidity theorems, Ergodic Theory Dyn. Syst., Volume 16 (1996) no. 4, pp. 623-649 | DOI | MR | Zbl

[8] Bourdon, Marc Sur le birapport au bord des CAT(-1)-espaces, Publ. Math., Inst. Hautes Étud. Sci., Volume 83 (1996), pp. 95-104 | DOI | Zbl

[9] Bowen, Rufus Topological entropy and axiom A, Global Analysis (Proceedings of Symposia in Pure Mathematics), Volume 14, American Mathematical Society, 1970, pp. 23-41 | DOI | Zbl

[10] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, 319, Springer, 2013

[11] Burger, Marc; Mozes, Shahar CAT (-1)-spaces, divergence groups and their commensurators, J. Am. Math. Soc., Volume 9 (1996) no. 1, pp. 57-93 | DOI | MR | Zbl

[12] Burns, Keith Hyperbolic behaviour of geodesic flows on manifolds with no focal points, Ph. D. Thesis, University of Wawrick (1983)

[13] Burns, Keith; Katok, Anatole; Ballman, W.; Brin, Michael I.; Eberlein, P.; Osserman, R. Manifolds with non-positive curvature, Ergodic Theory Dyn. Syst., Volume 5 (1985) no. 2, pp. 307-317 | DOI | Zbl

[14] Butt, Karen Approximate control of the marked length spectrum by short geodesics (2022) | arXiv

[15] Butt, Karen Quantitative marked length spectrum rigidity (2022) (to appear in Geometry & Topology)

[16] Butt, Karen Closed geodesics and stability of negatively curved metrics, Ph. D. Thesis, University of Michigan (2023)

[17] Cantrell, Stephen; Reyes, Eduardo Marked length spectrum rigidity from rigidity on subsets (2023) | arXiv

[18] Croke, Christopher B. Rigidity for surfaces of non-positive curvature, Comment. Math. Helv., Volume 65 (1990) no. 1, pp. 150-169 | DOI | Zbl

[19] Croke, Christopher B.; Dairbekov, Nurlan S. Lengths and volumes in Riemannian manifolds, Duke Math. J., Volume 125 (2004) no. 1, pp. 1-14 | MR | Zbl

[20] Croke, Christopher B.; Fathi, Albert; Feldman, Joel The marked length-spectrum of a surface of nonpositive curvature, Topology, Volume 31 (1992) no. 4, pp. 847-855 | DOI | MR | Zbl

[21] Douady, Adrien; Earle, Clifford J Conformally natural extension of homeomorphisms of the circle, Acta Math., Volume 157 (1986), pp. 23-48 | DOI | MR | Zbl

[22] Duistermaat, J. J.; Guillemin, V. W. The Spectrum of Positive Elliptic Operators and Periodic Bicharacteristics, Invent. Math., Volume 29 (1975), pp. 39-79 | DOI | MR | Zbl

[23] Farb, Benson; Margalit, Dan A primer on mapping class groups, Princeton University Press, 2011 | DOI

[24] Farrell, F. Thomas; Jones, Lowell E. Negatively curved manifolds with exotic smooth structures, J. Am. Math. Soc., Volume 2 (1989) no. 4, pp. 899-908 | DOI | MR | Zbl

[25] Farrell, F. Thomas; Jones, Lowell E. A topological analogue of Mostow’s rigidity theorem, J. Am. Math. Soc., Volume 2 (1989) no. 2, pp. 257-370 | MR | Zbl

[26] Farrell, F. Thomas; Jones, Lowell E. Complex hyperbolic manifolds and exotic smooth structures, Invent. Math., Volume 117 (1994) no. 1, pp. 57-74 | DOI | MR | Zbl

[27] Farrell, F. Thomas; Ontaneda, Pedro On the topology of the space of negatively curved metrics, J. Differ. Geom., Volume 86 (2010) no. 2, pp. 273-302 | MR | Zbl

[28] Feres, Renato The minimal entropy theorem and Mostow rigidity: after G. Besson, G. Courtois and S. Gallot (1996) (unpublished)

[29] Fisher, Todd; Hasselblatt, Boris Hyperbolic flows, European Mathematical Society, 2019 | DOI

[30] Fricke, Robert; Klein, Felix Vorlesungen über die Theorie der elliptischen Modulfunctionen, BG Teubner, 1890

[31] Furstenberg, Harry A Poisson formula for semi-simple Lie groups, Ann. Math. (1963), pp. 335-386 | DOI | MR | Zbl

[32] Gogolev, Andrey; Rodriguez Hertz, Federico Smooth rigidity for 3-dimensional volume preserving Anosov flows and weighted marked length spectrum rigidity (2022) | arXiv

[33] Gordon, Carolyn Survey of isospectral manifolds, Handbook of Differential Geometry, North-Holland, 1999, pp. 747-778 | Zbl

[34] Gouëzel, Sébastien; Lefeuvre, Thibault Classical and microlocal analysis of the X-ray transform on Anosov manifolds, Anal. PDE, Volume 14 (2021) no. 1, pp. 301-322 | DOI | MR | Zbl

[35] Gromov, Mikhaıl Three remarks on geodesic dynamics and fundamental group, Enseign. Math., Volume 46 (2000) no. 3-4, pp. 391-402 | MR | Zbl

[36] Guillarmou, Colin; Knieper, Gerhard; Lefeuvre, Thibault Geodesic stretch, pressure metric and marked length spectrum rigidity, Ergodic Theory Dyn. Syst., Volume 42 (2022) no. 3, pp. 974-1022 | DOI | MR | Zbl

[37] Guillarmou, Colin; Lefeuvre, Thibault The marked length spectrum of Anosov manifolds, Ann. Math., Volume 190 (2019) no. 1, pp. 321-344 | MR | Zbl

[38] Guillarmou, Colin; Lefeuvre, Thibault; Paternain, Gabriel P. Marked length spectrum rigidity for Anosov surfaces (2023) | arXiv

[39] Guillemin, Victor; Kazhdan, David Some inverse spectral results for negatively curved 2-manifolds, Topology, Volume 19 (1980) no. 3, pp. 301-312 | DOI | MR | Zbl

[40] Hamenstädt, Ursula Time-preserving conjugacies of geodesic flows, Ergodic Theory Dyn. Syst., Volume 12 (1992) no. 1, pp. 67-74 | DOI | MR | Zbl

[41] Hamenstädt, Ursula Regularity of time-preserving conjugacies for contact Anosov flows with C1-Anosov splitting, Ergodic Theory Dyn. Syst., Volume 13 (1993) no. 1, pp. 65-72 | DOI | MR | Zbl

[42] Hamenstädt, Ursula Cocycles, Hausdorff measures and cross ratios, Ergodic Theory Dyn. Syst., Volume 17 (1997) no. 5, pp. 1061-1081 | DOI | MR | Zbl

[43] Hamenstädt, Ursula Cocycles, symplectic structures and intersection, Geom. Funct. Anal., Volume 9 (1999) no. 1, pp. 90-140 | DOI | MR | Zbl

[44] Hasselblatt, Boris Horospheric foliations and relative pinching, J. Differ. Geom., Volume 39 (1994) no. 1, pp. 57-63 | MR | Zbl

[45] Hersonsky, Sa’ar; Paulin, Frédéric On the rigidity of discrete isometry groups of negatively curved spaces, Comment. Math. Helv., Volume 72 (1997) no. 3, pp. 349-388 | DOI | MR | Zbl

[46] Hirsch, Morris W.; Pugh, Charles C. Smoothness of horocycle foliations, J. Differ. Geom., Volume 10 (1975) no. 2, pp. 225-238 | MR | Zbl

[47] Huber, Heinz Zur analytischen theorie hyperbolischer raumformen und bewegungsgruppen, Math. Ann., Volume 138 (1959) no. 1, pp. 1-26 | DOI | Zbl

[48] Kac, Mark Can one hear the shape of a drum?, Am. Math. Mon., Volume 73 (1966) no. 4P2, pp. 1-23 | MR | Zbl

[49] Katok, Anatole Entropy and closed geodesics, Ergodic Theory Dyn. Syst., Volume 2 (1982) no. 3-4, pp. 339-365 | DOI | MR | Zbl

[50] Katok, Anatole; Hasselblatt, Boris Introduction to the modern theory of dynamical systems, Cambridge University Press, 1997 no. 54

[51] Katok, Svetlana Approximate solutions of cohomological equations associated with some Anosov flows, Ergodic Theory Dyn. Syst., Volume 10 (1990) no. 2, pp. 367-379 | DOI | MR | Zbl

[52] Knieper, Gerhard Hyperbolic dynamics and Riemannian geometry, Handbook of dynamical systems, Volume 1, Elsevier, 2002, pp. 453-545 | Zbl

[53] Livšic, A. N. Cohomology of dynamical systems, Math. USSR, Izv., Volume 6 (1972) no. 6, p. 1278 | DOI

[54] Manning, Anthony Topological entropy for geodesic flows, Ann. Math., Volume 110 (1979) no. 3, pp. 567-573 | DOI | MR | Zbl

[55] Margulis, Grigoriy A. Applications of ergodic theory to the investigation of manifolds of negative curvature, Funkts. Anal. Prilozh., Volume 3 (1969) no. 4, pp. 89-90 | MR

[56] Margulis, Grigoriy A. On some aspects of the theory of Anosov systems, Springer, 2004 | DOI | MR

[57] McKean, Henry P. Selberg’s trace formula as applied to a compact Riemann surface, Commun. Pure Appl. Math., Volume 25 (1972) no. 3, pp. 225-246 | DOI | MR

[58] Milnor, John Eigenvalues of the Laplace operator on certain manifolds, Proc. Natl. Acad. Sci. USA, Volume 51 (1964) no. 4, p. 542-542 | DOI | MR | Zbl

[59] Mostow, G. Daniel Strong Rigidity of Locally Symmetric Spaces, University of Tokyo Press, 1973 no. 78

[60] Otal, Jean-Pierre Le spectre marqué des longueurs des surfaces à courbure négative, Ann. Math., Volume 131 (1990) no. 1, pp. 151-162 | DOI | Zbl

[61] Otal, Jean-Pierre Sur la géométrie symplectique de l’espace des géodésiques d’une variété à courbure négative, Rev. Mat. Iberoam., Volume 8 (1992) no. 3, pp. 441-456 | DOI | MR | Zbl

[62] Ruan, Yuping The Cayley hyperbolic space and volume entropy rigidity (2022) | arXiv

[63] Sawyer, Noelle Partial marked length spectrum rigidity for negatively curved surfaces, Ph. D. Thesis, Wesleyan University (2020)

[64] Schmutz, Paul Die parametrisierung des Teichmüllerraumes durch geodätische Längenfunktionen, Comment. Math. Helv., Volume 68 (1993) no. 1, pp. 278-288 | DOI | MR | Zbl

[65] Sigmund, Karl On the space of invariant measures for hyperbolic flows, Am. J. Math., Volume 94 (1972) no. 1, pp. 31-37 | DOI | MR | Zbl

[66] Song, Antoine Entropy and stability of hyperbolic metrics (2023) | arXiv

[67] Sunada, Toshikazu Riemannian coverings and isospectral manifolds, Ann. Math., Volume 121 (1985) no. 1, pp. 169-186 | DOI | MR | Zbl

[68] Thurston, William P. Minimal stretch maps between hyperbolic surfaces (1998) | arXiv

[69] Colin de Verdière, Yves Spectre du laplacien et longueurs de géodesiques périodiques I, Compos. Math., Volume 27 (1973) no. 1, pp. 83-106 | Zbl

[70] Colin de Verdière, Yves Spectre du laplacien et longueurs de géodesiques périodiques II, Compos. Math., Volume 27 (1973) no. 2, pp. 159-184 | Zbl

[71] Vignéras, Marie-France Variétés riemanniennes isospectrales et non isométriques, Ann. Math., Volume 112 (1980) no. 1, pp. 21-32 | DOI | Zbl

[72] Wilkinson, Amie Lectures on marked length spectrum rigidity (preliminary version) (2012) (http://www.math.utah.edu/pcmi12/lecture_notes/wilkinson.pdf)

[73] Zelditch, Steve The inverse spectral problem, Surveys in differential geometry. Eigenvalues of Laplacians and other geometric operators (Surveys in Differential Geometry), Volume 9, International Press, 2004, pp. 401-467 | DOI | MR | Zbl