Abstract
We establish, as an application of the results from Eliashberg and Murphy (Lagrangian caps, 2013), an h-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space \({\mathbb{R}^6_{\rm st}}\) with exactly one transverse double point. Our construction also yields a Lagrangian embedding \({S^1 \times S^2 \to \mathbb{R}^6_{\rm st}}\) with vanishing Maslov class.
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References
Asadi-Golmankhaneh M., Eccles P.: Double point self-intersection surfaces of immersions. Geometry and Topology, 4, 149–170 (2000)
Audin M. (1988) Fibrés normaux d’immersions en dimension double, points doubles d’immersions lagragiennes et plongements totalement réels. Commentarii Mathematici Helvetici 63(4): 593–623
K. Cieliebak and Y. Eliashberg. From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds. In: American Mathematical Society Colloquium Publications, Vol. 59. American Mathematical Society, Providence (2012).
M. Damian. Floer homology on the universal cover, a proof of Audin’s conjecture, and other constraints on Lagrangian submanifolds. Commentarii Mathematici Helvetici, 87 (2012), 433–462.
T. Ekholm, J. Etnyre, and J. Sabloff. A duality exact sequence for Legendrian contact homology. Duke Mathematical Journal, (1)150 (2009), 1–75
T. Ekholm, J. Etnyre, and M. Sullivan. Non-isotopic Legendrian submanifolds in \({\mathbb{R}^{2n+1}}\) . Journal of Differential Geometry, 71 (2005), 85–128.
Ekholm T., Etnyre J., Sullivan M.: Orientations in Legendrian contact homology and exact Lagrangian immersions.. International Journal of Mathematics, 5(16), 453–532 (2005)
T. Ekholm and I. Smith. Exact Lagrangian Immersions with a Single Double Point. Preprint, arXiv:1111.5932.
T. Ekholm and I. Smith. Exact Lagrangian Immersions with One Double Point Revisited. Preprint, arXiv:1211.1715.
Y. Eliashberg. Topological characterization of Stein manifolds of dimension > 2. International Journal of Mathematics 1 (1990), 29–46.
Y. Eliashberg and M. Gromov. Lagrangian intersection theory: finite-dimensional approach. In: Geometry of differential equations, American Mathematical Society Translations Ser. 2, Vol. 186. (1998), pp. 27–118.
Y. Eliashberg and E. Murphy. Lagrangian Caps. Geometric and Functional Analysis (2013). doi:10.1007/s00039-013-1239-2.
K. Fukaya. Application of Floer homology of Lagrangian submanifolds to symplectic topology. In: Morse theoretic methods in nonlinear analysis and in symplectic topology, NATO Science Series II, Mathematics, Physics and Chemistry, Vol. 217. Springer, Berlin (2006), pp. 231–276.
K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono. Lagrangian intersection Floer theory: anomaly and obstruction. Part I. American Mathematical Society, International Press (2009).
M. Gromov. Partial differential relations. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, Ser. 3, Vol. 9. Springer, Berlin (1986).
M. Gromov. Pseudoholomorphic curves in symplectic manifolds. Inventiones Mathematicae, (2)82 (1985), 307–347.
E. Murphy. Loose Legendrian Embeddings in High Dimensional Contact Manifolds. Preprint, arXiv:1201.2245.
L. Polterovich. The surgery of Lagrange submanifolds. Journal of Functional Analysis, 2 (1991), 198–210.
D. Sauvaget. Curiosités lagrangiennes en dimension 4. Annales de l’institut Fourier (Grenoble), (6)54 (2004), 1997–2020.
C. Viterbo. A new obstruction to embedding Lagrangian tori. Inventiones Mathematicae, 100 (1990), 301–320.
Whitney H. (1944) The self-intersections of a smooth n-manifold in 2n-space. Annals of Mathematics 45(2): 220–246
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T. Ekholm is partially supported by Swedish Research Council Grant 2012-2365 and by the Knut and Alice Wallenberg Foundation as a Wallenberg Scholar.
Y. Eliashberg is partially supported by NSF grant DMS-1205349.
E. Murphy is partially supported by NSF grant DMS-0943787.
I. Smith is partially supported by grant ERC-2007-StG-205349 from the European Research Council.
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Ekholm, T., Eliashberg, Y., Murphy, E. et al. Constructing exact Lagrangian immersions with few double points. Geom. Funct. Anal. 23, 1772–1803 (2013). https://doi.org/10.1007/s00039-013-0243-6
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DOI: https://doi.org/10.1007/s00039-013-0243-6