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Transactions of the American Mathematical Society

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Marked-length-spectral rigidity for flat metrics

Authors: Anja Bankovic and Christopher J. Leininger
Journal: Trans. Amer. Math. Soc. 370 (2018), 1867-1884
MSC (2010): Primary 57M50
Published electronically: October 31, 2017
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Abstract: In this paper we prove that the space of flat metrics (nonpositively curved Euclidean cone metrics) on a closed, oriented surface is marked-length-spectrally rigid. In other words, two flat metrics assigning the same lengths to all closed curves differ by an isometry isotopic to the identity. The novel proof suggests a stronger rigidity conjecture for this class of metrics.

References [Enhancements On Off] (What's this?)

  • [AKP] S. Alexander, V. Kapovitch, and A. Petrunin,
    Alexandrov geometry,
    preprint,$ \sim $sba/, 2014.
  • [AL13] Javier Aramayona and Christopher J. Leininger, Hyperbolic structures on surfaces and geodesic currents, Algorithmic and Geometric Topics Around Free Groups and Automorphisms, Birkhäuser, 2017.
  • [Ban14] Anja Bankovic, Horowitz-Randol pairs of curves in $ q$-differential metrics, Algebr. Geom. Topol. 14 (2014), no. 5, 3107-3139. MR 3276858,
  • [BH99] Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486
  • [Bon86] Francis Bonahon, Bouts des variétés hyperboliques de dimension $ 3$, Ann. of Math. (2) 124 (1986), no. 1, 71-158 (French). MR 847953,
  • [Bon88] Francis Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), no. 1, 139-162. MR 931208,
  • [BS85] Joan S. Birman and Caroline Series, Geodesics with bounded intersection number on surfaces are sparsely distributed, Topology 24 (1985), no. 2, 217-225. MR 793185,
  • [CFF92] C. Croke, A. Fathi, and J. Feldman, The marked length-spectrum of a surface of nonpositive curvature, Topology 31 (1992), no. 4, 847-855. MR 1191384,
  • [Con15] David Constantine, Marked length spectrum rigidity in nonpositive curvature with singularities,
    preprint, arXiv:1507.04970, 2015.
  • [Cro90] Christopher B. Croke, Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv. 65 (1990), no. 1, 150-169. MR 1036134,
  • [Dan14] Klaus Dankwart, Rigidity of flat surfaces under the boundary measure, Israel J. Math. 199 (2014), no. 2, 623-640. MR 3219551,
  • [DLR10] Moon Duchin, Christopher J. Leininger, and Kasra Rafi, Length spectra and degeneration of flat metrics, Invent. Math. 182 (2010), no. 2, 231-277. MR 2729268,
  • [FK65] Robert Fricke and Felix Klein, Vorlesungen über die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen. Band II: Die funktionentheoretischen Ausführungen und die Andwendungen, Bibliotheca Mathematica Teubneriana, Bände 3, vol. 4, Johnson Reprint Corp., New York; B. G. Teubner Verlagsgesellschaft, Stuttg art, 1965 (German). MR 0183872
  • [Fra12] Jeffrey Frazier, Length spectral rigidity of non-positively curved surfaces, ProQuest LLC, Ann Arbor, MI, 2012. Thesis (Ph.D.)-University of Maryland, College Park. MR 3067870
  • [HP97] Sa'ar Hersonsky and Frédéric Paulin, On the rigidity of discrete isometry groups of negatively curved spaces, Comment. Math. Helv. 72 (1997), no. 3, 349-388. MR 1476054,
  • [MS91] Howard Masur and John Smillie, Hausdorff dimension of sets of nonergodic measured foliations, Ann. of Math. (2) 134 (1991), no. 3, 455-543. MR 1135877,
  • [Ota90] Jean-Pierre Otal, Le spectre marqué des longueurs des surfaces à courbure négative, Ann. of Math. (2) 131 (1990), no. 1, 151-162 (French). MR 1038361,
  • [Pat99] Gabriel P. Paternain, Geodesic flows, Progress in Mathematics, vol. 180, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1712465
  • [Res] Yu. G. Reshetnyak, Non-expansive maps in a space of curvature no greater than $ K$, Sibirsk. Mat. Ž. 9 (1968), 918-927; English transl., Siberian Math. J. 9 (1968), 683-687 (Russian). MR 0244922

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Additional Information

Anja Bankovic
Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02459
Address at time of publication: Cara Lazara 2/6, 34220 Laovo, Serbia

Christopher J. Leininger
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

Received by editor(s): July 16, 2015
Received by editor(s) in revised form: June 16, 2016
Published electronically: October 31, 2017
Additional Notes: The second author was partially supported by NSF grants DMS-1207183 and DMS-1510034
Article copyright: © Copyright 2017 American Mathematical Society

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