Skip to main content
SpringerLink
Log in

Super Stable Kählerian Horseshoe?

  • Chapter
  • First Online:
Essays in Mathematics and its Applications

Abstract

Following the prints of Smale’s horseshoe, we trace the problems originated from the interface between hyperbolic stability and the Abel-Jacobi-Albanese construction.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature. J. Differ. Geom. 18, 701–721 (1983)

    Google Scholar 

  2. D. Anosov, Geodesic Flows on Closed Riemannian Manifolds with Negative Curvature. Proceedings of the Steklov institute of mathematics, vol. 90 (American Mathematical Society, Providence, 1967)

    Google Scholar 

  3. G. Band, Identifying points of pseudo-Anosov homeomorphisms. Fund. Math. 180(2), 185–198 (2003)

    Google Scholar 

  4. M. Barge, J. Kwapisz, Hyperbolic pseudo-Anosov maps a. e. embed into a toral automorpism. Ergod. Theory Dyn. Syst. 264, 961–972 (2006). http://www.math.montana.edu/~jarek/Papers/hirsh.pdf

  5. R. Bowen, Markov partitions for axiom A diffeomorphisms. Am. J. Math. 91, 725–747 (1970)

    Google Scholar 

  6. M. Brunnbauer, D. Kotschick, On hyperbolic cohomology classes (2008). arxiv.org/pdf/0808.1482

  7. J. Carlson, D. Toledo, Harmonic mappings of Kähler manifolds to locally symmetric spaces. Inst. Hautes Études Sci. Publ. Math. 69, 173–201 (1989)

    Google Scholar 

  8. J. Carlson, D. Toledo, Rigidity of harmonic maps of maximum rank. J. Geom. Anal. 32, 99–140 (1993)

    Google Scholar 

  9. K. Corlette, Harmonic maps, rigidity, and Hodge theory, in Proceedings of the International Congress of Mathematicians (Zürich, 1994), vol. 1 (Birkhäuser, Basel, 1995), pp. 465–471

    Google Scholar 

  10. G. Daskalopoulos, C. Mese, A. Vdovina, Superrigidity of hyperbolic buildings. Geom. Funct. Anal. 1–15 (2011). Springer

    Google Scholar 

  11. G. Daskalopoulos, L. Katzarkov, R. Wentworth, Harmonic maps to Teichmuller space. Math. Res. Lett. 71, 133–146 (2000)

    Google Scholar 

  12. T. Delzant, L’invariant de Bieri Neumann Strebel des groupes fondamentaux des variétés kähleriennes. Math. Ann. 348, 119–125 (2010)

    Google Scholar 

  13. T. Delzant, M. Gromov, Cuts in Kähler Groups. Progress in mathematics, vol. 248 (Birkhäuser, Basel/Switzerland, 2005), pp. 31–55

    Google Scholar 

  14. T. Delzant. P. Py, Kähler groups, real hyperbolic spaces and the Cremona group (2010). arXiv:1012.1585v1

    Google Scholar 

  15. S. Donaldson, Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. 55:1(3), 127–131 (1987)

    Google Scholar 

  16. A. Eskin, D. Fisher, K. Whyte, Quasi-isometries and rigidity of solvable groups. Pure Appl. Math. Q. 3(4), 927–947 (2007) (Special Issue: In honor of Gregory Margulis)

    Google Scholar 

  17. A. Fathi, Homotopical stability of pseudo-Anosov diffeomorphisms. Erg. Theory Dyn. Syst. 10, 287–294 (1990)

    Google Scholar 

  18. D. Fisher, Local rigidity: past, present, future, in Dynamics, Ergodic Theory and Geometry. Mathematical sciences research institute publications (Cambridge University Press, Cambridge/New York, 2007), p. 45–98. A p.s file available on http://mypage.iu.edu/~fisherdm/paperwork.html

  19. D. Fisher, T. Hitchman, Cocycle superrigidity and harmonic maps with infinite dimensional targets (2005). arXiv:math/0511666

    Google Scholar 

  20. D. Fisher, K. Whyte, Continuous quotients for lattice actions on compact manifolds. Joint with Kevin Whyte. Geome. Ded. 87, 181–189 2001. A p.s file available on http://mypage.iu.edu/~fisherdm/paperwork.html

  21. J. Franks, Anosov Diffeomorphisms. Proceedings of symposia in pure mathematics, vol. 14 (American Mathematical Society, Providence, 1970), pp. 61–93

    Google Scholar 

  22. J. Franks, M. Handel, Complete semi-conjugacies for psuedo-Anosov homeomorphisms. (2007). http://www.math.wisc.edu/\(sim\)oh/FranksHandel.pdf

  23. D. Fried, Finitely presented dynamical systems. Erg. Theory Dyn. Syst. 7, 489–507 (1987)

    Google Scholar 

  24. A. Gournay. A Runge approximation theorem for pseudo-holomorphic maps (2010). arXiv:1006.2005v1

    Google Scholar 

  25. H. Grauert, Über modifikationen und exzeptionelle analytische Mengen. Math. Ann. 146(4), 331–368 (1962)

    Google Scholar 

  26. M. Gromov, Hyperbolic manifolds, groups and actions, in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference. Annals of mathematics studies, vol. 97 (Princeton University Press, Princeton, 1981) pp. 183–213

    Google Scholar 

  27. M. Gromov, Volume and Bounded Cohomology. Institut des hautes études scientifiques (Paris, France).; Publications mathématiques, vol. 56 ( I.H.E.S, Le Bois-Marie, Bures s/Yvette, 1982/1983), pp. 5–99

    Google Scholar 

  28. M. Gromov, Hyperbolic groups, in Essays in Group Theory, Mathematical sciences research institute publications, vol. 8 (Springer, New York, 1987) pp. 75–263

    Google Scholar 

  29. M. Gromov, Foliated plateau problem, part I. GAFA 1 14–79 (1991)

    Google Scholar 

  30. M. Gromov, Foliated plateau problem, part II. GAFA 1 253–320 (1991)

    Google Scholar 

  31. M. Gromov, Kähler hyperbolicity and \({L}_{2}\)-Hodge theory. J. Differ. Geom. 33(1), 263–292 (1991)

    Google Scholar 

  32. M. Gromov. Stability and pinching, in Seminari di Geometria, ed. by M. Ferri (Universitàdegli studi di Bologna, Dipartimento di matematica, Bologna, 1992), pp. 55–99

    Google Scholar 

  33. M. Gromov, Asymptotic Invariants of Infinite Groups. Geometric group theory (Sussex, 1991), vol. 2, London mathematical society lecture note series, vol. 182 (Cambridge University Press, Cambridge, 1993), pp. 1–295

    Google Scholar 

  34. M. Gromov. Positive curvature, macroscopic dimension, spectral gaps and higher signatures, in Functional Analysis on the Eve of the 21st Century, Volume II, ed. by S. Gindikin et al., (In honor of the eightieth birthday of I. M. Gelfand). Proceedings conference, Rutgers University, New Brunswick, 24–27 October 1993. Progress in mathematics, vol. 132 (Birkhäuser, Basel, 1996), pp. 1–213

    Google Scholar 

  35. M. Gromov, Carnot-CarathŽodory spaces seen from within, in Sub-Riemannian Geometry. Progress in mathematics, vol. 144, (Birkhäuser, Basel, 1996), pp. 79–323

    Google Scholar 

  36. M. Gromov, Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. 1(2), 109–197 (1999)

    Google Scholar 

  37. M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps: I. Math. Phys. Anal. Geom. 2 323–415 (1999)

    Google Scholar 

  38. M. Gromov, Spaces and questions. Geom. Funct. Anal. Special Volume(Part I), 118–161 (2000). GAFA 2000

    Google Scholar 

  39. M. Gromov, Random walk in random groups. GAFA, Geom. Funct. Anal. 13, 73–146 (2003)

    Google Scholar 

  40. M. Gromov, Manifolds: where do we come from ? What are we? Where are we going? (2010)

    Google Scholar 

  41. M. Gromov, R. Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Publ. Math. IHES 76, 165–246 (1992)

    Google Scholar 

  42. M. Gromov, G. Henkin, M. Shubin, Holomorphic L2 functions on coverings of pseudoconvex manifolds. Geom. Funct. Anal. 8(3), 552–585 (1998)

    Google Scholar 

  43. J. Hadamard, Les surfaces à courbures opposèes et leurs lignes gèodèsique. J. Math. Pures Appl. 4, 27–73 (1898)

    Google Scholar 

  44. L. Hernandez, Kähler manifolds and 1/4-pinching. Duke Math. J. 62(3), 601–611 (1991)

    Google Scholar 

  45. M. Hirsch, On invariant subsets of hyperbolic sets, in Essays on Topology and Related Topics (Springer, New York, 1970), pp. 126–135

    Google Scholar 

  46. S. Iitaka, Logarithmic forms of algebraic varieties, J. Fac. Sci. 23, 525–544 (1976)

    Google Scholar 

  47. Y. Ishii, J. Smillie, Homotopy shadowing. Am. J. Math. 132(4), 987–1029 (2010)

    Google Scholar 

  48. J.Jost, Y. Yang, Kähler manifolds and fundamental groups of negatively \(\delta \)-pinched manifolds. arXiv:math/0312143v1

    Google Scholar 

  49. J. Jost, S.-T. Yau, Harmonic maps and superrigidity. Proc. Symp. Pure Math. 54, 245–280 (1993)

    Google Scholar 

  50. J. Jost, S.-T. Yau, A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry. Acta Math. 170(2), 221–254 (1993). Errata in 173 (1994), 307

    Google Scholar 

  51. J. Jost, K. Zuo, Harmonic maps of infinite energy and rigidity results for representations of fundamental groups of quasiprojective varieties. Math. Res. Lett. 1, 631–638 (1994)

    Google Scholar 

  52. V. Kaimanovich, H. Masur, The Poisson boundary of the mapping class group. Invent. Math. 125(2), 221–264 (1996)

    Google Scholar 

  53. A. Katok, J. Lewis, R. Zimmer, Cocycle superrigidity and rigidity for lattice actions on tori. Topology 35, 27–38 (1996)

    Google Scholar 

  54. B. Kleiner, B. Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings. Inst. Hautes tudes Sci. Publ. Math. 86, 115–197 (1997)

    Google Scholar 

  55. N. Korevaar, R. Schoen, Sobolev spaces and harmonic maps for metric space targets, Commun. Anal. Geom. 1(3–4), 561–659 (1993)

    Google Scholar 

  56. F. Labourie, ÒExistence d’applications harmoniques tordues a valeurs dans les variétes á courbure negativeÒ. Proc. Am. Math. Soc. 111(3), 877–882 (1991)

    Google Scholar 

  57. G. Larotonda, Geodesic Convexity Symmetric Spaces and Hilbert-Schmidt Operators Gabriel Larotonda, These, 2004, Instituto de Ciencias Universidad Nacional de General Sarmiento JM Gutirrez 1150 (1613) Los Polvorines Buenos Aires, Argentina

    Google Scholar 

  58. J. Lohkamp, An existence theorem for harmonic maps. Manuscr. Math. 67(1), 21–23 (1990)

    Google Scholar 

  59. G. Lusztig, Cohomology of classifying spaces and Hermitian representations. Represent. Theory Electron. J. Am. Math. Soc. 1, 31–36 (1996)

    Google Scholar 

  60. G. Margulis, N. Qian, Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices. Ergodic Theory Dyn. Syst. 21, 121–164 (2001)

    Google Scholar 

  61. S. Matsuo, M. Tsukamoto, Instanton approximation, periodic ASD connections, and mean dimension. arXiv:0909.1141

    Google Scholar 

  62. I. Mineyev, N. Monod, Y. Shalom, Ideal bicombings for hyperbolic groups and applications. arXiv.org ¿ math ¿ arXiv:math/0304278

    Google Scholar 

  63. N. Mok, Harmonic forms with values in locally constant Hilbert bundles, in Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993). J. Fourier Anal. Appl. (Special Issue), 433–453

    Google Scholar 

  64. N. Mok, Y. Siu, S. Yeung, Geometric superrigidity. Invent. Math. 113, 57–83 (1993)

    Google Scholar 

  65. H. M. Morse, A one-to-one representation of geodesics on a surface of negative curvature. Am. J. Math. 43(1), 33–51 (1921). JSTOR

    Google Scholar 

  66. D. Mumford, Rational equivalence of 0-cycles on surfaces. J. Math. Kyoto Univ. 9, 195–204 (1969)

    Google Scholar 

  67. V. Nekrashevych, Symbolic dynamics and self-similar groups (Preprint).

    Google Scholar 

  68. P. Pansu, Mtriques de Carnot-Carathodory et quasiisomtries des espaces symtriques de rang un. J. Ann. Math. 129(1), 1–60 (1989)

    Google Scholar 

  69. B. Popescu, Infinite dimensional symmetric spaces Inaugural-Dissertation zur Erlangung des Doktorgrades der Matematisch-Naturwissenschaftlichen Fakultat der Universitat Augsburg vorgelegt von Augsburg 2005

    Google Scholar 

  70. Th. M. Rassias, Isometries and approximate isometries. Int. J. Math. Math. Sci. 25, 73–91 (2001)

    Google Scholar 

  71. E. Rips, Subgroups of small cancellation groups. Bull. Lond. Math. Soc. 14(1), 45–47 (1982)

    Google Scholar 

  72. J. Sampson, Applications of harmonic maps to Kähler geometry, in Complex Differential Geometry and Nonlinear Differential Equations, Brunswick, 1984. Contemporary Mathematics, vol. 49 (American Mathematical Society, Providence, 1986), pp. 125–134

    Google Scholar 

  73. B. Schmidt, Weakly Hyperbolic Actions of Kazhdan Groups on Tori. arXiv:math/0511314

    Google Scholar 

  74. G. Schumacher, Harmonic maps of the moduli space of compact Riemann surfaces. Math. Ann. 275(3), 455–466 (1986)

    Google Scholar 

  75. J.-P. Serre, Groupes algebriques et corps dc classes. Actualités scientifiques et industielles (Hermann, Paris, 1959)

    Google Scholar 

  76. M. Shub, Endomorphisms of compact differentiable manifolds. Am. J. Math. XCI, 175–199 (1969)

    Google Scholar 

  77. Y. Sinai, Markov partitions and C-diffeomorphisms. Funct. Anal. Appl. 2(1), 64–89 (1968). Funkts. Anal. Prilozh. 2(1), 64–89 (1968)

    Google Scholar 

  78. Y.T. Siu, The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. Ann. Math. (2) 112(1), 73–111 (1980)

    Google Scholar 

  79. S. Smale, Finding a horseshoe on the beaches of Rio. Math. Intell. 20(1), 39–44 (1998)

    Google Scholar 

  80. D. Toledo, Rigidity theorems in Kähler geometry and fundamental groups of varieties, in Several Complex Variables. MSRI Publications, (Berkeley, CA, 1995–1996), vol. 37 (Cambridge University Press, Cambridge, UK/New York, 1999), pp. 509–533

    Google Scholar 

  81. D. Toledo, Energy is Plurisubharmonic. http://www.math.utah.edu/~toledo/energy.pdf

  82. C. Voisin, On the homotopy types of Kähler manifolds and the birational Kodaira problem. J. Differ. Geom. 72, 43–71 (2006)

    Google Scholar 

  83. M. Wolf, Infinite energy harmonic maps and degeneration of hyperbolic surfaces in moduli spaces. J. Differ. Geom. 33, 487–539 (1991)

    Google Scholar 

  84. K. Wortman, A finitely presented solvable group with small quasi-isometry group. Mich. Math. J. 55, 3–24 (2007)

    Google Scholar 

  85. R. Zimmer, Actions of semisimple groups and discrete subgroups, in Proceedings of the International Congress of Mathematicians, Berkeley 1247–1258 (1986)

    Google Scholar 

Download references

Acknowledgements

I thank Domingo Toledo who acquainted me with his unpublished \(\mathbb{C}\)-convexity result and also suggested several corrections and improvement in the present manuscript. I am also thankful to John Franks and Jarek Kwapisz who instructed me on the (relatively) recent development in the theory of hyperbolic, in particular pseudo-Anosov, systems associated with B.

Author information

Authors and Affiliations

  1. IHES, Bur-sur-Yvette, France

    M. Gromov

  2. CIMS, New York, USA

    M. Gromov

Authors
  1. M. Gromov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Gromov .

Editor information

Editors and Affiliations

  1. , Department of Industrial & Systems Engin, University of Florida, Weil Hall 401, Gainesville, 32611, Florida, USA

    Panos M. Pardalos

  2. , Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens, 15780, Greece

    Themistocles M. Rassias

Additional information

Dedicated to the 80th Anniversary of Professor Stephen Smale

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Gromov, M. (2012). Super Stable Kählerian Horseshoe?. In: Pardalos, P., Rassias, T. (eds) Essays in Mathematics and its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28821-0_8

Download citation

Publish with us

Policies and ethics

Search

Navigation