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The minimal length of a Lagrangian cobordism between Legendrians

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Abstract

To investigate the rigidity and flexibility of Lagrangian cobordisms between Legendrian submanifolds, we study the minimal length of such a cobordism, which is a 1-dimensional measurement of the non-cylindrical portion of the cobordism. Our primary tool is a set of real-valued capacities for a Legendrian submanifold, which are derived from a filtered version of Legendrian contact homology. Relationships between capacities of Legendrians at the ends of a Lagrangian cobordism yield lower bounds on the length of the cobordism. We apply the capacities to Lagrangian cobordisms realizing vertical dilations (which may be arbitrarily short) and contractions (whose lengths are bounded below). We also study the interaction between length and the linking of multiple cobordisms as well as the lengths of cobordisms derived from non-trivial loops of Legendrian isotopies.

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References

  1. Audin, M., Lalonde, F., Polterovich, L.: Symplectic Rigidity: Lagrangian submanifolds, Holomorphic Curves in Symplectic Geometry, Progress in Mathematics, vol. 117, pp. 271–321. Birkhäuser, Basel (1994)

  2. Bourgeois, F., Chantraine, B.: Bilinearized Legendrian contact homology and the augmentation category. J. Symplectic Geom. 12(3), 553–583 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Bourgeois, F., Sabloff, J., Traynor, L.: Lagrangian cobordisms via generating families: construction and geography. Algebr. Geom. Topol. 15(4), 2439–2477 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chantraine, B.: On Lagrangian concordance of Legendrian knots. Algebr. Geom. Topol. 10, 63–85 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chantraine, B.: A note on exact Lagrangian cobordisms with disconnected Legendrian ends. Proc. Am. Math. Soc. 143(3), 1325–1331 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chantraine, B., Dimitroglou Rizell, G., Ghiggini, P., Golovko, R.: Floer homology and Lagrangian concordance. In: Proceedings of the Gökova Geometry–Topology Conference 2014, pp. 76–113 .Gökova Geometry/Topology Conference (GGT), Gökova (2015)

  8. Chantraine, B., Dimitroglou Rizell, G., Ghiggini, P., Golovko, R.: Floer theory for Lagrangian cobordisms. (2015). Preprint available as arXiv:1511.09471

  9. Chekanov, Yu.: Differential algebra of Legendrian links. Invent. Math. 150, 441–483 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Civan, G., Etnyre, J., Koprowski, P., Sabloff, J., Walker, A.: Product structures for Legendrian contact homology. Math. Proc. Camb. Philos. Soc. 150(2), 291–311 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dimitroglou Rizell, G.: Legendrian ambient surgery and Legendrian contact homology. (2012). Preprint available as arXiv:1205.5544v1

  12. Dimitroglou Rizell, G.: Lifting pseudo-holomorphic polygons to the symplectisation of \(P\times \mathbb{R}\) and applications. Quantum Topol. 7(1), 29–105 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ekholm, T.: Rational symplectic field theory over \(\mathbb{Z}_2\) for exact Lagrangian cobordisms. J. Eur. Math. Soc. (JEMS) 10(3), 641–704 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ekholm, T.: Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology. In: Perspectives in Analysis, Geometry, and Topology, Progress in Mathematics, vol. 296, pp. 109–145. Birkhäuser/Springer, New York (2012)

  15. Ekholm, T., Etnyre, J., Sabloff, J.: A duality exact sequence for Legendrian contact homology. Duke Math. J. 150(1), 1–75 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ekholm, T., Etnyre, J., Sullivan, M.: The contact homology of Legendrian submanifolds in \({\mathbb{R}}^{2n+1}\). J. Differential Geom. 71(2), 177–305 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ekholm, T., Etnyre, J., Sullivan, M.: Non-isotopic Legendrian submanifolds in \(\mathbb{R}^{2n+1}\). J. Differential Geom. 71(1), 85–128 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ekholm, T., Etnyre, J., Sullivan, M.: Legendrian contact homology in \(P\times \mathbb{R}\). Trans. Am. Math. Soc 359(7), 3301–3335 (2007). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ekholm, T., Honda, K., Kálmán, T.: Legendrian knots and exact Lagrangian cobordisms. (2012). Preprint available as arXiv:1212.1519

  20. Eliashberg, Y.: Invariants in contact topology. iN: Proceedings of the International Congress of Mathematicians, vol. II (Berlin, 1998), no. extra vol. II, pp. 327–338 (1998) (electronic)

  21. Eliashberg, Ya., Givental, A., Hofer, H.: Introduction to symplectic field theory. Geom. Funct. Anal. (no. Special Volume, Part II), 560–673 (2000)

  22. Eliashberg Ya., Gromov, M.: Lagrangian intersection theory: finite-dimensional approach. In: Geometry of Differential Equations, American Mathematical Society Translations: Series 2, vol. 186, pp. 27–118. American Mathematical Society, Providence, RI (1998)

  23. Etnyre, J.: Legendrian and Transversal Knots, Handbook of Knot Theory. Elsevier B. V, Amsterdam (2005)

    MATH  Google Scholar 

  24. Fuchs, D.: Chekanov–Eliashberg invariant of Legendrian knots: existence of augmentations. J. Geom. Phys. 47(1), 43–65 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fuchs, D., Ishkhanov, T.: Invariants of Legendrian knots and decompositions of front diagrams. Mosc. Math. J. 4(3), 707–717 (2004)

    MathSciNet  MATH  Google Scholar 

  26. Fukaya, K., Oh, Y-G., Ohta, H., Ono, K.: Lagrangian Intersection Floer Theory: Anomaly and Obstruction. Parts I and II, AMS/IP Studies in Advanced Mathematics, vol. 46. American Mathematical Society, Providence, RI (2009)

  27. Golovko, R.: A note on Lagrangian cobordisms between Legendrian submanifolds of \(\mathbb{R}^{2n+1}\). Pac. J. Math. 261(1), 101–116 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Golovko, R.: A note on the front spinning construction. Bull. Lond. Math. Soc. 46(2), 258–268 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hutchings, M.: Quantitative embedded contact homology. J. Differ. Geom. 88(2), 231–266 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kadeišvili, T.: On the theory of homology of fiber spaces, Uspekhi Mat. Nauk 35 (1980), no. 3(213), 183–188, International Topology Conference (Moscow State University, Moscow, 1979)

  32. Kálmán, T.: Contact homology and one parameter families of Legendrian knots. Geom. Topol. 9, 2013–2078 (2005). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  33. Melvin, P., Shrestha, S.: The nonuniqueness of Chekanov polynomials of Legendrian knots. Geom. Topol. 9, 1221–1252 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mishachev, K.: The \(n\)-copy of a topologically trivial Legendrian knot. J. Symplectic Geom. 1(4), 659–682 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ng, L., Rutherford, D., Shende, V., Sivek, S., and Zaslow, E.:Augmentations are sheaves. (2015). Preprint available as arXiv:1502.04939

  36. Sabloff, J.: Augmentations and rulings of Legendrian knots. Int. Math. Res. Notes 19, 1157–1180 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sabloff, J.: Duality for Legendrian contact homology. Geom. Topol. 10, 2351–2381 (2006). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  38. Sabloff, J., Sullivan, M.: Families of Legendrian submanifolds via generating families. Quantum Topol. (to appear). arXiv:1311.0528

  39. Sabloff, J., Traynor, L.: Obstructions to the existence and squeezing of Lagrangian cobordisms. J. Topol. Anal. 2(2), 203–232 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Sabloff, J., Traynor, L.: Obstructions to Lagrangian cobordisms between Legendrian submanifolds. Algebr. Geom. Topol. 13, 2733–2797 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  41. Seidel, Paul: Fukaya categories and Picard–Lefschetz theory. European Mathematical Society (EMS), Zürich, Zurich Lectures in Advanced Mathematics (2008)

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

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Haverford College, Haverford, PA, 19041, USA

    Joshua M. Sabloff

  2. Bryn Mawr College, Bryn Mawr, PA, 19010, USA

    Lisa Traynor

Authors
  1. Joshua M. Sabloff
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  2. Lisa Traynor
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Corresponding author

Correspondence to Joshua M. Sabloff.

Additional information

JS is partially supported by NSF Grant DMS-1406093. LT gratefully acknowledges the hospitality of the Institute for Advanced Study in Princeton and support at IAS from The Fund for Mathematics during a portion of this work.

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Sabloff, J.M., Traynor, L. The minimal length of a Lagrangian cobordism between Legendrians. Sel. Math. New Ser. 23, 1419–1448 (2017). https://doi.org/10.1007/s00029-016-0288-0

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  • DOI: https://doi.org/10.1007/s00029-016-0288-0

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