Abstract
We make the elementary observation that the Lagrangian submanifolds of C n, n≥3, constructed by Ekholm, Eliashberg, Murphy and Smith are non-uniruled and, moreover, have infinite relative Gromov width. The construction of these submanifolds involve exact Lagrangian caps, which obviously are non-uniruled in themselves. This property is also used to show that if a Legendrian submanifold inside a contactisation admits an exact Lagrangian cap, then its Chekanov–Eliashberg algebra is acyclic.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barraud, J.-F. and Cornea, O., Homotopic dynamics in symplectic topology, in Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, NATO Sci. Ser. II Math. Phys. Chem. 217, pp. 109–148, Springer, Dordrecht, 2006.
Barraud, J.-F. and Cornea, O., Lagrangian intersections and the Serre spectral sequence, Ann. of Math. 166 (2007), 657–722.
Biran, P. and Cornea, O., Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol. 13 (2009), 2881–2989.
Borman, M. S. and McLean, M., Bounding Lagrangian widths via geodesic paths, to appear in Compos. Math.
Charette, F., A geometric refinement of a theorem of Chekanov, J. Symplectic Geom. 10 (2012), 475–491.
Chekanov, Y., Differential algebra of Legendrian links, Invent. Math. 150 (2002), 441–483.
Cornea, O. and Lalonde, F., Cluster homology, Preprint, 2005. arXiv:math/0508345.
Dimitroglou Rizell, G., Legendrian ambient surgery and Legendrian contact homology, Preprint, 2012. arXiv:1205.5544.
Dimitroglou Rizell, G., Knotted Legendrian surfaces with few Reeb chords, Algebr. Geom. Topol. 11 (2011), 2903–2936.
Ekholm, T., Eliashberg, Y., Murphy, E. and Smith, I., Constructing exact Lagrangian immersions with few double points, Geom. Funct. Anal. 23 (2013), 1772–1803.
Ekholm, T., Etnyre, J. B. and Sabloff, J. M., A duality exact sequence for Legendrian contact homology, Duke Math. J. 150 (2009), 1–75.
Ekholm, T., Etnyre, J. and Sullivan, M., Non-isotopic Legendrian submanifolds in R 2n+1, J. Differential Geom. 71 (2005), 85–128.
Ekholm, T., Etnyre, J. and Sullivan, M., The contact homology of Legendrian submanifolds in R 2n+1, J. Differential Geom. 71 (2005), 177–305.
Ekholm, T., Etnyre, J. and Sullivan, M., Orientations in Legendrian contact homology and exact Lagrangian immersions, Internat. J. Math. 16 (2005), 453–532.
Ekholm, T., Etnyre, J. and Sullivan, M., Legendrian contact homology in P×R, Trans. Amer. Math. Soc. 359 (2007), 3301–3335.
Ekholm, T., Honda, K. and Kálmán, T., Legendrian knots and exact Lagrangian cobordisms, Preprint, 2012. arXiv:1212.1519.
Ekholm, T. and Kálmán, T., Isotopies of Legendrian 1-knots and Legendrian 2-tori, J. Symplectic Geom. 6 (2008), 407–460.
Eliashberg, Y., Givental, A. and Hofer, H., Introduction to symplectic field theory, Geom. Funct. Anal., Special Volume, Part II (2000), 560–673.
Eliashberg, Y. and Murphy, E., Lagrangian caps, Geom. Funct. Anal. 23 (2013), 1483–1514.
Fukaya, K., Application of Floer homology of Lagrangian submanifolds to symplectic topology, in Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, NATO Sci. Ser. II Math. Phys. Chem. 217, pp. 231–276, Springer, Dordrecht, 2006.
Golovko, R., A note on the front spinning construction, Bull. Lond. Math. Soc. 46 (2014), 258–268.
Gromov, M., Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347.
Gromov, M., Partial Differential Relations, Ergebnisse der Mathematik und ihrer Grenzgebiete 9, Springer, Berlin, 1986.
Lalonde, F. and Sikorav, J.-C., Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents, Comment. Math. Helv. 66 (1991), 18–33.
Lees, J. A., On the classification of Lagrange immersions, Duke Math. J. 43 (1976), 217–224.
Lin, F., Exact Lagrangian caps of Legendrian knots, Preprint, 2013. arXiv:1309.5101.
Murphy, E., Loose Legendrian embeddings in high dimensional contact manifolds, Preprint, 2012. arXiv:1201.2245.
Murphy, E., Closed exact Lagrangians in the symplectization of contact manifolds, Preprint, 2013. arXiv:1304.6620.
Ng, L., Computable Legendrian invariants, Topology 42 (2003), 55–82.
Polterovich, L., The surgery of Lagrange submanifolds, Geom. Funct. Anal. 1 (1991), 198–210.
Sauvaget, D., Curiosités lagrangiennes en dimension 4, Ann. Inst. Fourier (Grenoble) 54 (2004), 1997–2020.
Sikorav, J.-C., Some properties of holomorphic curves in almost complex manifolds, in Holomorphic Curves in Symplectic Geometry, Progr. Math. 117, pp. 165–189, Birkhäuser, Basel, 1994.
Zehmisch, K., The codisc radius capacity, Electron. Res. Announc. Math. Sci. 20 (2013), 77–96.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by the ERC starting grant of Frédéric Bourgeois StG-239781-ContactMath.
Rights and permissions
About this article
Cite this article
Dimitroglou Rizell, G. Exact Lagrangian caps and non-uniruled Lagrangian submanifolds. Ark Mat 53, 37–64 (2015). https://doi.org/10.1007/s11512-014-0202-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11512-014-0202-y