Skip to main content

Viterbo conjecture for Zoll symmetric spaces

Abstract

We prove a conjecture of Viterbo from 2007 on the existence of a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in unit cotangent disk bundles, for bases given by compact rank one symmetric spaces \(S^n, {\mathbb {R}}P^n, {\mathbb {C}}P^n, {\mathbb {H}}P^n,\) \(n\ge 1.\) We discuss generalizations and give applications, in particular to \(C^0\) symplectic topology. Our key method consists in a quantitative deformation argument for Floer persistence modules that allows to excise a divisor.

This is a preview of subscription content, access via your institution.

Notes

  1. Save for the isolated case of the Cayley plane \({\mathbb {O}}P^2,\) that will be treated separately elsewhere.

  2. I thank the referees for bringing this point to my attention.

  3. Note that \(\mu _L\) comes from a class in \(H^2(M,L;{\mathbb {Z}}).\)

  4. Namely, \(\widetilde{\mathcal {L}}^{\min }_{pt} M = \widetilde{\mathcal {L}}_{pt}M \times (2 N_M \cdot {\mathbb {Z}})/ \pi _1(\mathcal {L}_{pt}M),\) where \(\alpha \in \pi _1(\mathcal {L}_{pt}M)\) acts on \((\eta ,2N_M m) \in \widetilde{\mathcal {L}}_{pt}M \times (2 N_M \cdot {\mathbb {Z}})\) by \(\alpha \cdot (\eta , 2N_M m) = (\alpha ^{-1} * \eta , 2N_M m + c_M(\alpha )),\) where \(*\) denotes the action induced by concatenation. This is the covering of \({\mathcal {L}}_{pt} M\) corresponding to \(\ker (c_M) \subset \pi _1(\mathcal {L}_{pt}M),\) with fibers explicitly identified with \({{\,\mathrm{\mathrm {im}}\,}}(c_M)\).

  5. This argument is inspired by conversations with D. Tonkonog and R. Vianna.

References

  1. Abouzaid, M.: Nearby Lagrangians with vanishing Maslov class are homotopy equivalent. Invent. Math. 189(2), 251–313 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Abouzaid, M., Kragh, T.: Simple homotopy equivalence of nearby Lagrangians. Acta Math. 220(2), 207–237 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Alvarez-Gavela, D., Kaminker, V., Kislev, A., Kliakhandler, K., Pavlichenko, A., Rigolli, L., Rosen, D., Shabtai, O., Stevenson, B., Zhang, J.: Embeddings of free groups into asymptotic cones of Hamiltonian diffeomorphisms. J. Topol. Anal. 11(2), 467–498 (2019)

    MathSciNet  MATH  Google Scholar 

  4. Arnol’d, V.I.: A stability problem and ergodic properties of classical dynamical systems. In: Proc. Internat. Congr. Math. (Moscow), pages 387–392 (1966)

  5. Arnol’d, V.I.: Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian. Uspehi Mat. Nauk 18((5 (113))), 13–40 (1963)

    MathSciNet  Google Scholar 

  6. Arnol’d, V.I.: Small denominators and problems of stability of motion in classical and celestial mechanics. Uspehi Mat. Nauk 18(6(114)), 91–192 (1963)

    MathSciNet  MATH  Google Scholar 

  7. Arnol’d, V.I.: Sur une propriété topologique des applications globalement canoniques de la mécanique classique. C. R. Acad. Sci. Paris 261, 3719–3722 (1965)

    MathSciNet  MATH  Google Scholar 

  8. Arnol’d, V.I.: Mathematical methods of classical mechanics, volume 60 of Graduate Texts in Mathematics. Springer-Verlag, New York,: Translated from the 1974 Russian original by K. Vogtmann and A, Weinstein (1989)

  9. Arnol’d, V.I.: Some remarks on symplectic monodromy of Milnor fibrations. In: The Floer memorial volume, volume 133 of Progr. Math., pages 99–103. Birkhäuser, Basel, (1995)

  10. Audin, M.: Lagrangian skeletons, periodic geodesic flows and symplectic cuttings. Manuscripta Math. 124(4), 533–550 (2007)

    MathSciNet  MATH  Google Scholar 

  11. Auroux, D., Gayet, D., Mohsen, J.-P.: Symplectic hypersurfaces in the complement of an isotropic submanifold. Math. Ann. 321(4), 739–754 (2001)

    MathSciNet  MATH  Google Scholar 

  12. Bates, S., Weinstein, A.: Lectures on the geometry of quantization, volume 8 of Berkeley Mathematics Lecture Notes. American Mathematical Society, Providence, RI; Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, (1997)

  13. Bauer, U., Lesnick, M.: Induced matchings and the algebraic stability of persistence barcodes. J. Comput. Geom. 6(2), 162–191 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Besse, A.L.: Manifolds all of whose geodesics are closed, volume 93 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]. Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan

  15. Biran, P.: Lagrangian barriers and symplectic embeddings. Geom. Funct. Anal. 11(3), 407–464 (2001)

    MathSciNet  MATH  Google Scholar 

  16. Biran, P.: Symplectic topology and algebraic families. In: European Congress of Mathematics, pages 827–836. Eur. Math. Soc., Zürich, (2005)

  17. Biran, P.: Lagrangian non-intersections. Geom. Funct. Anal. 16(2), 279–326 (2006)

    MathSciNet  MATH  Google Scholar 

  18. Biran, P., Cornea, O.: Bounds on the Lagrangian spectral metric in cotangent bundles. Comment. Math. Helv. 96(4), 631–691 (2021)

    MathSciNet  MATH  Google Scholar 

  19. Biran, P., Cornea, O.: Quantum structures for Lagrangian submanifolds. Preprint arXiv:0708.4221 (2007)

  20. Biran, P., Cornea, O.: A Lagrangian quantum homology. In: New perspectives and challenges in symplectic field theory, volume 49 of CRM Proc. Lecture Notes, pp. 1–44. Amer. Math. Soc, Providence, RI (2009)

  21. Biran, P., Cornea, O.: Rigidity and uniruling for Lagrangian submanifolds. Geom. Topol. 13(5), 2881–2989 (2009)

    MathSciNet  MATH  Google Scholar 

  22. Biran, P., Cornea, O.: Lagrangian topology and enumerative geometry. Geom. Topol. 16(2), 963–1052 (2012)

    MathSciNet  MATH  Google Scholar 

  23. Biran, P., Cornea, O.: Lagrangian cobordism I. J. Amer. Math. Soc. 26(2), 295–340 (2013)

    MathSciNet  MATH  Google Scholar 

  24. Biran, P., Cornea, O., Shelukhin, E.: Lagrangian shadows and triangulated categories. Astérisque, (426):128, (2021)

  25. Biran, P., Entov, M., Polterovich, L.: Calabi quasimorphisms for the symplectic ball. Commun. Contemp. Math. 6(5), 793–802 (2004)

    MathSciNet  MATH  Google Scholar 

  26. Bourgeois, F., Eliashberg, Y., Hofer, H., Wysocki, K., Zehnder, E.: Compactness results in symplectic field theory. Geom. Topol. 7, 799–888 (2003)

    MathSciNet  MATH  Google Scholar 

  27. Brandenbursky, M., Kȩdra, J., Shelukhin, E.: On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus. Commun. Contemp. Math. 20(2), 1750042, 27 (2018)

    MathSciNet  MATH  Google Scholar 

  28. Buhovsky, L., Humilière, V., Seyfaddini, S.: An Arnold-type principle for non-smooth objects. J. Fixed Point Theory Appl. 24(2), 24 (2022)

  29. Buhovsky, L., Humilière, V., Seyfaddini, S.: The action spectrum and \(C^0\) symplectic topology. Math. Ann. 380(1–2), 293–316 (2021)

    MathSciNet  MATH  Google Scholar 

  30. Calegari, D.: scl. Mathematical Society of Japan, Tokyo (2009)

    MATH  Google Scholar 

  31. Carlsson, G., Zomorodian, A.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005)

    MathSciNet  MATH  Google Scholar 

  32. Chaperon, M.: Une idée du type géodésiques brisées pour les systèmes hamiltoniens. C. R. Acad. Sci. Paris Sér. I Math. 298(13), 293–296 (1984)

    MathSciNet  MATH  Google Scholar 

  33. Charest, F., Woodward, C.: Floer theory and flips. Mem. Amer. Math. Soc., to appear. Available at arXiv:1508.01573

  34. Charest, F., Woodward, C.: Floer trajectories and stabilizing divisors. J. fixed point theory appl. 19(2), 1165–1236 (2017)

    MathSciNet  MATH  Google Scholar 

  35. Charette, F.: A geometric refinement of a theorem of Chekanov. J. Symplectic Geom. 10(3), 475–491 (2012)

    MathSciNet  MATH  Google Scholar 

  36. Chazal, F., de Silva, V., Glisse, M., Oudot, S.: The structure and stability of persistence modules. SpringerBriefs in Mathematics, Springer, [Cham] (2016)

    MATH  Google Scholar 

  37. Chazal, F., Steiner, D.C., Glisse, M., Guibas, L.J., Oudot, S.Y.: Proximity of persistence modules and their diagrams. In: Proceedings of the 25th Annual Symposium on Computational Geometry, SCG ’09, pages 237–246. ACM, (2009)

  38. Chekanov, Y.V.: Invariant Finsler metrics on the space of Lagrangian embeddings. Math. Z. 234(3), 605–619 (2000)

    MathSciNet  MATH  Google Scholar 

  39. Cieliebak, K., Mohnke, K.: Compactness for punctured holomorphic curves. J. Symplectic Geom. 3(4), 589–654 (2005). (Conference on Symplectic Topology)

    MathSciNet  MATH  Google Scholar 

  40. Cieliebak, K., Mohnke, K.: Symplectic hypersurfaces and transversality in Gromov-Witten theory. J. Symplectic Geom. 5(3), 281–356 (2007)

    MathSciNet  MATH  Google Scholar 

  41. Cieliebak, K., Mohnke, K.: Punctured holomorphic curves and Lagrangian embeddings. Invent. Math. 212(1), 213–295 (2018)

    MathSciNet  MATH  Google Scholar 

  42. Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37(1), 103–120 (2007)

    MathSciNet  MATH  Google Scholar 

  43. Conley, C.C., Zehnder, E.: The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol’d. Invent. Math. 73(1), 33–49 (1983)

    MathSciNet  MATH  Google Scholar 

  44. Crawley-Boevey, W.: Decomposition of pointwise finite-dimensional persistence modules. J. Algebra Appl. 14(5), 1550066, 8 (2015)

    MathSciNet  MATH  Google Scholar 

  45. Dimitroglou Rizell, G., Sullivan, M.: The persistence of the Chekanov-Eliashberg algebra. Selecta Math. (N.S.) 26(5), 69, 32 (2020)

    MathSciNet  MATH  Google Scholar 

  46. Entov, M.: Quasi-morphisms and quasi-states in symplectic topology. In: Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, pages 1147–1171. Kyung Moon Sa, Seoul, (2014)

  47. Entov, M., Polterovich, L.: Calabi quasimorphism and quantum homology. Int. Math. Res. Not. 30, 1635–1676 (2003)

    MathSciNet  MATH  Google Scholar 

  48. Entov, M., Polterovich, L., Py, P.: On continuity of quasimorphisms for symplectic maps. In: Perspectives in analysis, geometry, and topology, volume 296 of Progr. Math., pages 169–197. Birkhäuser/Springer, New York, 2012. With an appendix by Michael Khanevsky

  49. Entov, M., Polterovich, L., Zapolsky, F.: Quasi-morphisms and the Poisson bracket. Pure Appl. Math. Q., 3(4, Special Issue: In honor of Grigory Margulis. Part 1):1037–1055, (2007)

  50. Floer, A.: Proof of the Arnol’d conjecture for surfaces and generalizations to certain Kähler manifolds. Duke Math. J. 53(1), 1–32 (1986)

    MathSciNet  MATH  Google Scholar 

  51. Floer, A.: Morse theory for fixed points of symplectic diffeomorphisms. Bull. Amer. Math. Soc. (N.S.) 16(2), 279–281 (1987)

    MathSciNet  MATH  Google Scholar 

  52. Floer, A.: Symplectic fixed points and holomorphic spheres. Comm. Math. Phys. 120(4), 575–611 (1989)

    MathSciNet  MATH  Google Scholar 

  53. Floer, A.: Witten’s complex and infinite-dimensional Morse theory. J. Differential Geom. 30(1), 207–221 (1989)

    MathSciNet  MATH  Google Scholar 

  54. Fraser, M.: Contact spectral invariants and persistence. Preprint arXiv:1502.05979, (2015)

  55. Frauenfelder, U.: The Arnold-Givental conjecture and moment Floer homology. Int. Math. Res. Not. 42, 2179–2269 (2004)

    MathSciNet  MATH  Google Scholar 

  56. Frauenfelder, U.: Rabinowitz action functional on very negative line bundles. Habilitationsschrift, Münich (2008)

    Google Scholar 

  57. Frauenfelder, U., Schlenk, F.: Hamiltonian dynamics on convex symplectic manifolds. Israel J. Math. 159, 1–56 (2007)

    MathSciNet  MATH  Google Scholar 

  58. Fukaya, K., Seidel, P., Smith, I.: Exact Lagrangian submanifolds in simply-connected cotangent bundles. Invent. Math. 172(1), 1–27 (2008)

    MathSciNet  MATH  Google Scholar 

  59. Fukaya, K., Seidel, P., Smith, I.: The symplectic geometry of cotangent bundles from a categorical viewpoint. In: Homological mirror symmetry, volume 757 of Lecture Notes in Phys., pages 1–26. Springer, Berlin, (2009)

  60. Gambaudo, J.-M., Ghys, E.: Commutators and diffeomorphisms of surfaces. Ergodic Theory Dynam. Systems 24(5), 1591–1617 (2004)

    MathSciNet  MATH  Google Scholar 

  61. Ghys, E.: Knots and dynamics. In: International Congress of Mathematicians. Vol. I, pages 247–277. Eur. Math. Soc., Zürich, (2007)

  62. Ginzburg, V.: The Conley conjecture. Ann. of Math. 172, 1127–1180 (2010)

    Google Scholar 

  63. Ginzburg, V.L., Gürel, B.Z.: Hamiltonian pseudo-rotations of projective spaces. Invent. Math. 214(3), 1081–1130 (2018)

    MathSciNet  MATH  Google Scholar 

  64. Giroux, E.: Remarks on Donaldson’s symplectic submanifolds. Pure Appl. Math. Q. 13(3), 369–388 (2017)

    MathSciNet  MATH  Google Scholar 

  65. Givental’, A.B.: Nonlinear generalization of the Maslov index. In: Theory of singularities and its applications, volume 1 of Adv. Soviet Math., pages 71–103. Amer. Math. Soc., Providence, RI, (1990)

  66. Glasner, E., Weiss, B.: The topological Rohlin property and topological entropy. Amer. J. Math. 123(6), 1055–1070 (2001)

    MathSciNet  MATH  Google Scholar 

  67. Glasner, E., Weiss, B.: Topological groups with Rohlin properties. Colloq. Math. 110(1), 51–80 (2008)

    MathSciNet  MATH  Google Scholar 

  68. Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)

    MathSciNet  MATH  Google Scholar 

  69. Hind, R.: Lagrangian spheres in \(S^2\times S^2\). Geom. Funct. Anal. 14(2), 303–318 (2004)

    MathSciNet  MATH  Google Scholar 

  70. Hind, R.: Lagrangian unknottedness in Stein surfaces. Asian J. Math. 16(1), 1–36 (2012)

    MathSciNet  MATH  Google Scholar 

  71. Hofer, H.: On the topological properties of symplectic maps. Proc. Roy. Soc. Edinburgh Sect. A 115(1–2), 25–38 (1990)

    MathSciNet  MATH  Google Scholar 

  72. Hofer, H.: Estimates for the energy of a symplectic map. Comment. Math. Helv. 68(1), 48–72 (1993)

    MathSciNet  MATH  Google Scholar 

  73. Hofer, H., Zehnder, E.: Symplectic invariants and Hamiltonian dynamics. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel, (1994)

  74. Hörmander, L.: Fourier integral operators. I. Acta Math. 127(1–2), 79–183 (1971)

    MathSciNet  MATH  Google Scholar 

  75. Humilière, V., Leclercq, R., Seyfaddini, S.: Coisotropic rigidity and \(C^0\)-symplectic geometry. Duke Math. J. 164(4), 767–799 (2015)

    MathSciNet  MATH  Google Scholar 

  76. Kawamoto, Y.: Homogeneous quasimorphisms, \(C^0\)-topology and Lagrangian intersection. Comment. Math. Helv., to appear. Available at arXiv:2006.07844

  77. Kawamoto, Y.: On \(C^0\)-continuity of the spectral norm for symplectically non-aspherical manifolds. Int. Math. Res. Not. https://doi.org/10.1093/imrn/rnab206

  78. Khanevsky, M.: Hofer’s metric on the space of diameters. J. Topol. Anal. 1(4), 407–416 (2009)

    MathSciNet  MATH  Google Scholar 

  79. Kislev, A., Shelukhin, E.: Bounds on spectral norms and barcodes. Geom. Topol. 25(7), 3257–3350 (2021)

    MathSciNet  MATH  Google Scholar 

  80. Kolmogorov, A.N.: Théorie générale des systèmes dynamiques et mécanique classique. In: Proceedings of the International Congress of Mathematicians, Amsterdam, 1954, Vol. 1, pages 315–333. Erven P. Noordhoff N.V., Groningen; North-Holland Publishing Co., Amsterdam, (1957)

  81. Lalonde, F., McDuff, D.: The geometry of symplectic energy. Ann. of Math. (2) 141(2), 349–371 (1995)

    MathSciNet  MATH  Google Scholar 

  82. Lamotke, K.: The topology of complex projective varieties after S. Lefschetz. Topology 20(1), 15–51 (1981)

    MathSciNet  MATH  Google Scholar 

  83. Lanzat, S.: Quasi-morphisms and symplectic quasi-states for convex symplectic manifolds. Int. Math. Res. Not. 23, 5321–5365 (2013)

    MathSciNet  MATH  Google Scholar 

  84. Laudenbach, F., Sikorav, J.-C.: Persistance d’intersection avec la section nulle au cours d’une isotopie hamiltonienne dans un fibré cotangent. Invent. Math. 82(2), 349–357 (1985)

    MathSciNet  MATH  Google Scholar 

  85. Le Roux, F.: Six questions, a proposition and two pictures on Hofer distance for Hamiltonian diffeomorphisms on surfaces. In: Symplectic topology and measure preserving dynamical systems, volume 512 of Contemp. Math., pages 33–40. Amer. Math. Soc., Providence, RI, (2010)

  86. Le Roux, F., Seyfaddini, S., Viterbo, C.: Barcodes and area-preserving homeomorphisms. Geom. Topol. 25(6), 2713–2825 (2021)

    MathSciNet  MATH  Google Scholar 

  87. Leclercq, R., Zapolsky, F.: Spectral invariants for monotone Lagrangians. J. Topol. Anal. 10(3), 627–700 (2018)

    MathSciNet  MATH  Google Scholar 

  88. Lerman, E.: Symplectic cuts. Math. Res. Lett. 2(3), 247–258 (1995)

    MathSciNet  MATH  Google Scholar 

  89. Liu, G.: Associativity of quantum multiplication. Comm. Math. Phys. 191(2), 265–282 (1998)

    MathSciNet  MATH  Google Scholar 

  90. Mann, K., Wolff, M.: Rigidity of mapping class group actions on \(S^1\). Geom. Topol. 24(3), 1211–1223 (2020)

    MathSciNet  MATH  Google Scholar 

  91. McDuff, D., Salamon, D.: \(J\)-holomorphic curves and symplectic topology. 2nd ed., volume 52. Providence, RI: American Mathematical Society (AMS), 2nd ed. edition, (2012)

  92. Membrez, C., Opshtein, E.: \({C}^0\)-rigidity of Lagrangian submanifolds and punctured holomorphic discs in the cotangent bundle. Compos. Math. 157(11), 2433–2493 (2021)

    MathSciNet  MATH  Google Scholar 

  93. Milinković, D., Oh, Y.-G.: Floer homology as the stable Morse homology. J. Korean Math. Soc. 34(4), 1065–1087 (1997)

    MathSciNet  MATH  Google Scholar 

  94. Milinković, D., Oh, Y.-G.: Generating functions versus action functional. Stable Morse theory versus Floer theory. In: Geometry, topology, and dynamics (Montreal, PQ, 1995), volume 15 of CRM Proc. Lecture Notes, pp. 107–125. Amer. Math. Soc, Providence, RI (1998)

    Google Scholar 

  95. Monzner, A., Vichery, N., Zapolsky, F.: Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization. J. Mod. Dyn. 6(2), 205–249 (2012)

    MathSciNet  MATH  Google Scholar 

  96. Monzner, A., Zapolsky, F.: A comparison of symplectic homogenization and Calabi quasi-states. J. Topol. Anal. 3(3), 243–263 (2011)

    MathSciNet  MATH  Google Scholar 

  97. Moser, J.: On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962, 1–20 (1962)

    MathSciNet  MATH  Google Scholar 

  98. Oh, Y.-G.: Spectral invariants, analysis of the Floer moduli space, and geometry of the Hamiltonian diffeomorphism group. Duke Math. J. 130(2), 199–295 (2005)

    MathSciNet  MATH  Google Scholar 

  99. Oh, Y.-G.: Symplectic topology and Floer homology. Vols. 1 and 2, volume 27 and 28 of New Mathematical Monographs. Cambridge University Press, Cambridge, (2015). Symplectic geometry and pseudoholomorphic curves

  100. Piunikhin, S., Salamon, D., Schwarz, M.: Symplectic Floer-Donaldson theory and quantum cohomology. In: Contact and symplectic geometry (Cambridge, 1994), volume 8 of Publ. Newton Inst., pages 171–200. Cambridge Univ. Press, Cambridge, (1996)

  101. Polterovich, L.: The geometry of the group of symplectic diffeomorphisms. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2001)

    MATH  Google Scholar 

  102. Polterovich, L., Rosen, D.: Function theory on symplectic manifolds. CRM Monograph Series, vol. 34. American Mathematical Society, Providence, RI (2014)

    MATH  Google Scholar 

  103. Polterovich, L., Shelukhin, E.: Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules. Selecta Math. (N.S.) 22(1), 227–296 (2016)

    MathSciNet  MATH  Google Scholar 

  104. Polterovich, L., Shelukhin, E., Stojisavljević, V.: Persistence modules with operators in Morse and Floer theory. Mosc. Math. J. 17(4), 757–786 (2017)

    MathSciNet  MATH  Google Scholar 

  105. Pöschel, J.: Integrability of Hamiltonian systems on Cantor sets. Comm. Pure Appl. Math. 35(5), 653–696 (1982)

    MathSciNet  MATH  Google Scholar 

  106. Py, P.: Quasi-morphismes de Calabi et graphe de Reeb sur le tore. C. R. Math. Acad. Sci. Paris 343(5), 323–328 (2006)

    MathSciNet  MATH  Google Scholar 

  107. Rabinowitz, P.H.: Periodic solutions of Hamiltonian systems. Comm. Pure Appl. Math. 31(2), 157–184 (1978)

    MathSciNet  Google Scholar 

  108. Rabinowitz, P.H.: Periodic solutions of a Hamiltonian system on a prescribed energy surface. J. Differential Equations 33(3), 336–352 (1979)

    MathSciNet  MATH  Google Scholar 

  109. Ritter, A.F.: The Novikov theory for symplectic cohomology and exact Lagrangian embeddings. ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–Massachusetts Institute of Technology

  110. Ruan, Y., Tian, G.: A mathematical theory of quantum cohomology. Math. Res. Lett. 1(2), 269–278 (1994)

    MathSciNet  MATH  Google Scholar 

  111. Ruan, Y., Tian, G.: A mathematical theory of quantum cohomology. J. Differential Geom. 42(2), 259–367 (1995)

    MathSciNet  MATH  Google Scholar 

  112. Salamon, D., Zehnder, E.: KAM theory in configuration space. Comment. Math. Helv. 64(1), 84–132 (1989)

    MathSciNet  MATH  Google Scholar 

  113. Schwarz, M.: On the action spectrum for closed symplectically aspherical manifolds. Pacific J. Math. 193(2), 419–461 (2000)

    MathSciNet  MATH  Google Scholar 

  114. Seidel, P.: Symplectic Floer homology and the mapping class group. Pacific J. Math. 206(1), 219–229 (2002)

    MathSciNet  MATH  Google Scholar 

  115. Seidel, P.: Fukaya categories and Picard-Lefschetz theory. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, (2008)

  116. Seyfaddini, S.: Descent and \(C^0\)-rigidity of spectral invariants on monotone symplectic manifolds. J. Topol. Anal. 4(4), 481–498 (2012)

    MathSciNet  MATH  Google Scholar 

  117. Seyfaddini, S.: \(C^0\)-limits of Hamiltonian paths and the Oh-Schwarz spectral invariants. Int. Math. Res. Not. IMRN 21, 4920–4960 (2013)

    MathSciNet  MATH  Google Scholar 

  118. Seyfaddini, S.: The displaced disks problem via symplectic topology. C. R. Math. Acad. Sci. Paris 351(21–22), 841–843 (2013)

    MathSciNet  MATH  Google Scholar 

  119. Shelukhin, E.: Symplectic cohomology and a conjecture of Viterbo. Preprint, arXiv:1904.06798

  120. Sikorav, J.-C.: Problèmes d’intersections et de points fixes en géométrie hamiltonienne. Comment. Math. Helv. 62(1), 62–73 (1987)

    MathSciNet  MATH  Google Scholar 

  121. Smith, I.: Floer cohomology and pencils of quadrics. Invent. Math. 189(1), 149–250 (2012)

    MathSciNet  MATH  Google Scholar 

  122. Spanier, E.H.: Algebraic topology. McGraw-Hill, New York (1966)

    MATH  Google Scholar 

  123. Stevenson, B.: A quasi-isometric embedding into the group of Hamiltonian diffeomorphisms with Hofer’s metric. Israel J. Math. 223(1), 141–195 (2018)

    MathSciNet  MATH  Google Scholar 

  124. Stojisavljević, V., Zhang, J.: Persistence modules, symplectic Banach-Mazur distance and Riemannian metrics. Internat. J. Math. 32(7), 2150040, 76 (2021)

    MathSciNet  MATH  Google Scholar 

  125. Usher, M.: Symplectic Banach-Mazur distances between subsets of \(\mathbb{C}^n\). J. Topol. Anal. 14(1), 231–286 (2022)

    MathSciNet  MATH  Google Scholar 

  126. Usher, M.: The sharp energy-capacity inequality. Commun. Contemp. Math. 12(3), 457–473 (2010)

    MathSciNet  MATH  Google Scholar 

  127. Usher, M.: Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds. Israel J. Math. 184, 1–57 (2011)

    MathSciNet  MATH  Google Scholar 

  128. Usher, M.: Hofer’s metrics and boundary depth. Ann. Sci. Éc. Norm. Supér. (4) 46(1), 57–128 (2013). (2013)

    MathSciNet  MATH  Google Scholar 

  129. Usher, M., Zhang, J.: Persistent homology and Floer-Novikov theory. Geom. Topol. 20(6), 3333–3430 (2016)

    MathSciNet  MATH  Google Scholar 

  130. Vérine, A.: Bohr-Sommerfeld Lagrangian submanifolds as minima of convex functions. J. Symplectic Geom. 18(1), 333–353 (2020)

    MathSciNet  MATH  Google Scholar 

  131. Viterbo, C.: Symplectic homogenization. Preprint arXiv:0801.0206, (2014)

  132. Viterbo, C.: Symplectic topology as the geometry of generating functions. Math. Ann. 292(4), 685–710 (1992)

    MathSciNet  MATH  Google Scholar 

  133. Viterbo, C.: Functors and computations in Floer homology with applications. I. Geom. Funct. Anal. 9(5), 985–1033 (1999)

    MathSciNet  MATH  Google Scholar 

  134. Witten, E.: Two-dimensional gravity and intersection theory on moduli space. In: Surveys in differential geometry (Cambridge, MA, 1990), pp. 243–310. Lehigh Univ, Bethlehem, PA (1991)

    Google Scholar 

  135. Zapolsky, F.: The Lagrangian Floer-quantum-PSS package and canonical orientations in Floer theory. Preprint arXiv:1507.02253, (2015)

  136. Zhang, J.: \(p\)-cyclic persistent homology and Hofer distance. J. Symplectic Geom. 17(3), 857–927 (2019)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I thank Peter Albers, Paul Biran, Octav Cornea, Asaf Kislev, Leonid Polterovich, Vukašin Stojisavljević, Dmitry Tonkonog, Renato Vianna, and Frol Zapolsky for fruitful collaborations during which I learnt many of the tools that I apply in this paper. I thank Sobhan Seyfaddini and Georgios Dimitroglou Rizell for useful conversations. I thank the referees for very helpful comments and suggestions, which have improved the paper. This notably includes the strategy of the proof of Proposition 18, suggested by one of the referees, which improves the original quantitative deformation argument for barcodes used in this paper. It has led to a simplification of the proof of Theorem B. This work was initiated and was partially carried out during my stay at the Institute for Advanced Study, where I was supported by NSF grant No. DMS-1128155. It was partially written during visits to Tel Aviv University, and to Ruhr-Universität Bochum. I thank these institutions and Helmut Hofer, Leonid Polterovich, and Alberto Abbondandolo, for their warm hospitality. At the University of Montréal, I was supported by an NSERC Discovery Grant and by the Fonds de recherche du Québec - Nature et technologies.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Egor Shelukhin.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shelukhin, E. Viterbo conjecture for Zoll symmetric spaces. Invent. math. 230, 321–373 (2022). https://doi.org/10.1007/s00222-022-01124-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-022-01124-x