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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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)
Bourgeois, F., Chantraine, B.: Bilinearized Legendrian contact homology and the augmentation category. J. Symplectic Geom. 12(3), 553–583 (2014)
Bourgeois, F., Eliashberg, Ya., Hofer, H., Wysocki, K., Zehnder, E.: Compactness results in symplectic field theory. Geom. Topol. 7, 799–888 (2003)
Bourgeois, F., Sabloff, J., Traynor, L.: Lagrangian cobordisms via generating families: construction and geography. Algebr. Geom. Topol. 15(4), 2439–2477 (2015)
Chantraine, B.: On Lagrangian concordance of Legendrian knots. Algebr. Geom. Topol. 10, 63–85 (2010)
Chantraine, B.: A note on exact Lagrangian cobordisms with disconnected Legendrian ends. Proc. Am. Math. Soc. 143(3), 1325–1331 (2015)
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)
Chantraine, B., Dimitroglou Rizell, G., Ghiggini, P., Golovko, R.: Floer theory for Lagrangian cobordisms. (2015). Preprint available as arXiv:1511.09471
Chekanov, Yu.: Differential algebra of Legendrian links. Invent. Math. 150, 441–483 (2002)
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)
Dimitroglou Rizell, G.: Legendrian ambient surgery and Legendrian contact homology. (2012). Preprint available as arXiv:1205.5544v1
Dimitroglou Rizell, G.: Lifting pseudo-holomorphic polygons to the symplectisation of \(P\times \mathbb{R}\) and applications. Quantum Topol. 7(1), 29–105 (2016)
Ekholm, T.: Rational symplectic field theory over \(\mathbb{Z}_2\) for exact Lagrangian cobordisms. J. Eur. Math. Soc. (JEMS) 10(3), 641–704 (2008)
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)
Ekholm, T., Etnyre, J., Sabloff, J.: A duality exact sequence for Legendrian contact homology. Duke Math. J. 150(1), 1–75 (2009)
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)
Ekholm, T., Etnyre, J., Sullivan, M.: Non-isotopic Legendrian submanifolds in \(\mathbb{R}^{2n+1}\). J. Differential Geom. 71(1), 85–128 (2005)
Ekholm, T., Etnyre, J., Sullivan, M.: Legendrian contact homology in \(P\times \mathbb{R}\). Trans. Am. Math. Soc 359(7), 3301–3335 (2007). (electronic)
Ekholm, T., Honda, K., Kálmán, T.: Legendrian knots and exact Lagrangian cobordisms. (2012). Preprint available as arXiv:1212.1519
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)
Eliashberg, Ya., Givental, A., Hofer, H.: Introduction to symplectic field theory. Geom. Funct. Anal. (no. Special Volume, Part II), 560–673 (2000)
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)
Etnyre, J.: Legendrian and Transversal Knots, Handbook of Knot Theory. Elsevier B. V, Amsterdam (2005)
Fuchs, D.: Chekanov–Eliashberg invariant of Legendrian knots: existence of augmentations. J. Geom. Phys. 47(1), 43–65 (2003)
Fuchs, D., Ishkhanov, T.: Invariants of Legendrian knots and decompositions of front diagrams. Mosc. Math. J. 4(3), 707–717 (2004)
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)
Golovko, R.: A note on Lagrangian cobordisms between Legendrian submanifolds of \(\mathbb{R}^{2n+1}\). Pac. J. Math. 261(1), 101–116 (2013)
Golovko, R.: A note on the front spinning construction. Bull. Lond. Math. Soc. 46(2), 258–268 (2014)
Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)
Hutchings, M.: Quantitative embedded contact homology. J. Differ. Geom. 88(2), 231–266 (2011)
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)
Kálmán, T.: Contact homology and one parameter families of Legendrian knots. Geom. Topol. 9, 2013–2078 (2005). (electronic)
Melvin, P., Shrestha, S.: The nonuniqueness of Chekanov polynomials of Legendrian knots. Geom. Topol. 9, 1221–1252 (2005)
Mishachev, K.: The \(n\)-copy of a topologically trivial Legendrian knot. J. Symplectic Geom. 1(4), 659–682 (2003)
Ng, L., Rutherford, D., Shende, V., Sivek, S., and Zaslow, E.:Augmentations are sheaves. (2015). Preprint available as arXiv:1502.04939
Sabloff, J.: Augmentations and rulings of Legendrian knots. Int. Math. Res. Notes 19, 1157–1180 (2005)
Sabloff, J.: Duality for Legendrian contact homology. Geom. Topol. 10, 2351–2381 (2006). (electronic)
Sabloff, J., Sullivan, M.: Families of Legendrian submanifolds via generating families. Quantum Topol. (to appear). arXiv:1311.0528
Sabloff, J., Traynor, L.: Obstructions to the existence and squeezing of Lagrangian cobordisms. J. Topol. Anal. 2(2), 203–232 (2010)
Sabloff, J., Traynor, L.: Obstructions to Lagrangian cobordisms between Legendrian submanifolds. Algebr. Geom. Topol. 13, 2733–2797 (2013)
Seidel, Paul: Fukaya categories and Picard–Lefschetz theory. European Mathematical Society (EMS), Zürich, Zurich Lectures in Advanced Mathematics (2008)
Viterbo, C.: Symplectic topology as the geometry of generating functions. Math. Ann. 292(4), 685–710 (1992)
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-016-0288-0