Exact Lagrangian caps and non-uniruled Lagrangian submanifolds. (English) Zbl 1321.57035

Rizell, Georgios Dimitroglou

Ark. Mat. 53, No. 1, 37-64 (2015).

A smooth \((2n+1)\)-dimensional manifold \(Y\) together with the choice of a maximally non-integrable field of tangent hyperplanes, given as \(\ker\lambda\) for a fixed choice of one-form \(\lambda\), is a contact manifold viewed as the pair \((Y,\lambda)\), and \(\lambda\wedge(d\,\lambda)^n\) is a volume form on \(Y\). An \(n\)-dimensional submanifold \(\Lambda\subset (Y,\lambda)\) which is tangent to \(\ker\lambda\) is called a Legendrian submanifold. A \(2n\)-dimensional manifold \(X\) together with a closed non-degenerate two-form \(\omega\) is a symplectic manifold. \((X,\omega)\) is exact if \(\omega=d\alpha\) is exact. A properly embedded Lagrangian submanifold without boundary \(L_{\Lambda,\varnothing}\subset(\mathbb R\times Y,d(e^t\lambda))\) of the symplectization of \((Y,\lambda)\) satisfying the property that \(L_{\Lambda,\varnothing}\) coincides with the cylinder \((-\infty, N)\times\Lambda\) outside of a compact set, is called Lagrangian cap of a Legendrian submanifold \(\Lambda\subset (Y,\lambda)\). A properly embedded Lagrangian submanifold without boundary \(L_{\varnothing,\Lambda}\subset(X,d\alpha)\) that coincides with the cylinder \((N,\infty)\times\Lambda\subset[0,\infty)\times Y\) outside of a compact set, is called Lagrangian filling of a Legendrian submanifold \(\Lambda\subset (Y,\lambda)\). The concatenation of \(L_{\varnothing,\Lambda}\) and \(L_{\Lambda,\varnothing}\) is the closed Lagrangian submanifold \(L=(L_{\varnothing,\Lambda}\cap\{(t,x);\;t\leq N\})\cup(L_{\Lambda,\varnothing}\cap\{(t,x);\;t\geq N\})\subset(X, d\alpha)\). A Lagrangian immersion \(L\subset(X,\omega)\) is called displaceable if there exists a time-dependent Hamiltonian on \(X\) whose induced Hamiltonian flow \(\varphi^s: X\to X\) satisfies \(L\cap\varphi^1(L)=\varnothing\). The number \(w(L,X)=\sup\{\pi r^2\in[0,\infty);\;\mathcal{B}(X,L,r)\neq\varnothing\}\) is called the Gromov width of a Lagrangian submanifold \(L\subset(X,\omega)\), where \(\mathcal{B}(X,L,r)\) is the set of symplectic embeddings of \(B^{2n}(r)\) into \((X,\omega)\). A Lagrangian immersion \(L\subset(X,\omega)\) whose self-intersections consist of transverse double-points is said to be uniruled if there is some \(A>0\) for which, for any compatible almost complex structure \(J\) on \(X\) and a point \(x\in L\), there exists a non-constant \(J\)-holomorphic disc in \(X\) having boundary on \(L\), having a boundary point mapping to \(x\), and whose \(\omega\)-area is at most \(A\). Making use of exact Lagrangian caps, in [Geom. Funct. Anal. 23, No. 6, 1772–1803 (2013; Zbl 1283.53074)], T. Ekholm et al. constructed various Lagrangian submanifolds of \(\mathbb C^n\).
In this paper, the author shows that these submanifolds are non-uniruled and have infinite Gromov width. Also, it is proven that if \(L\subset(X,d\alpha)\), where \(\alpha\) is a primitive of symplectic form, is a displaceable exact Lagrangian immersion of a closed manifold obtained as the concatenation of an immersed exact Lagrangian filling and an embedded exact Lagrangian cap, then the Chekanov-Eliashberg algebra of \(L\) is acyclic even when using Novikov coefficients. Finally, the author considers a displaceable exact Lagrangian immersion \(L\subset(X,d\alpha)\) of a closed manifold with the assumption that \(L\) is a spin manifold with fixed choice of a spin structure. It is shown that if the Chekanov-Eliashberg algebra with coefficients in a unital ring \(R\) of the Legendrian lift of \(L\) is not acyclic, with or without Novikov coefficients, then for any \(J\in \mathcal{J}_L\) and \(x\in L\) there exists a \(J\)-holomorphic disc in \(X\) having boundary on \(L\), one positive boundary puncture, possibly several negative boundary punctures, and a boundary-point passing through \(x\), where \(\mathcal{J}_L\) is the set of compatible almost complex structures on \((X,d\alpha)\).

Reviewer: Andrew Bucki (Edmond)

MSC:

57R17	Symplectic and contact topology in high or arbitrary dimension
53D42	Symplectic field theory; contact homology

Keywords:

contact manifold; Legendrian submanifold; Lagrangian cap; Lagrangian filling; displaceable immersion; Gromov width; uniruled immersion

Citations:

Zbl 1283.53074

References:

[1]	Barraud, J.-F.; Cornea, O., Homotopic dynamics in symplectic topology, No. 217, 109-148 (2006), Dordrecht
[2]	Barraud, J.-F. and Cornea, O., Lagrangian intersections and the Serre spectral sequence, Ann. of Math.166 (2007), 657-722. · Zbl 1141.53078
[3]	Biran, P. and Cornea, O., Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol.13 (2009), 2881-2989. · Zbl 1180.53078
[4]	Borman, M. S. and McLean, M., Bounding Lagrangian widths via geodesic paths, to appear in Compos. Math. · Zbl 1322.53082
[5]	Charette, F., A geometric refinement of a theorem of Chekanov, J. Symplectic Geom.10 (2012), 475-491. · Zbl 1256.53050
[6]	Chekanov, Y., Differential algebra of Legendrian links, Invent. Math.150 (2002), 441-483. · Zbl 1029.57011
[7]	Cornea, O. and Lalonde, F., Cluster homology, Preprint, 2005. arXiv:math/0508345. · Zbl 1113.53052
[8]	Dimitroglou Rizell, G., Legendrian ambient surgery and Legendrian contact homology, Preprint, 2012. arXiv:1205.5544. · Zbl 1356.53078
[9]	Dimitroglou Rizell, G., Knotted Legendrian surfaces with few Reeb chords, Algebr. Geom. Topol.11 (2011), 2903-2936. · Zbl 1248.53073
[10]	Ekholm, T., Eliashberg, Y., Murphy, E. and Smith, I., Constructing exact Lagrangian immersions with few double points, Geom. Funct. Anal.23 (2013), 1772-1803. · Zbl 1283.53074
[11]	Ekholm, T., Etnyre, J. B. and Sabloff, J. M., A duality exact sequence for Legendrian contact homology, Duke Math. J.150 (2009), 1-75. · Zbl 1193.53179
[12]	Ekholm, T., Etnyre, J. and Sullivan, M., Non-isotopic Legendrian submanifolds in R2n+1, J. Differential Geom.71 (2005), 85-128. · Zbl 1098.57013
[13]	Ekholm, T., Etnyre, J. and Sullivan, M., The contact homology of Legendrian submanifolds in R2n+1, J. Differential Geom.71 (2005), 177-305. · Zbl 1103.53048
[14]	Ekholm, T., Etnyre, J. and Sullivan, M., Orientations in Legendrian contact homology and exact Lagrangian immersions, Internat. J. Math.16 (2005), 453-532. · Zbl 1076.53099
[15]	Ekholm, T., Etnyre, J. and Sullivan, M., Legendrian contact homology in P×R, Trans. Amer. Math. Soc.359 (2007), 3301-3335. · Zbl 1119.53051
[16]	Ekholm, T., Honda, K. and Kálmán, T., Legendrian knots and exact Lagrangian cobordisms, Preprint, 2012. arXiv:1212.1519. · Zbl 1357.57044
[17]	Ekholm, T. and Kálmán, T., Isotopies of Legendrian 1-knots and Legendrian 2-tori, J. Symplectic Geom.6 (2008), 407-460. · Zbl 1206.57030
[18]	Eliashberg, Y., Givental, A. and Hofer, H., Introduction to symplectic field theory, Geom. Funct. Anal., Special Volume, PartII (2000), 560-673. · Zbl 0989.81114
[19]	Eliashberg, Y. and Murphy, E., Lagrangian caps, Geom. Funct. Anal.23 (2013), 1483-1514. · Zbl 1308.53121
[20]	Fukaya, K., Application of Floer homology of Lagrangian submanifolds to symplectic topology, No. 217, 231-276 (2006), Dordrecht · Zbl 1089.53064
[21]	Golovko, R., A note on the front spinning construction, Bull. Lond. Math. Soc.46 (2014), 258-268. · Zbl 1287.53069
[22]	Gromov, M., Pseudoholomorphic curves in symplectic manifolds, Invent. Math.82 (1985), 307-347. · Zbl 0592.53025
[23]	Gromov, M., Partial Differential Relations, Ergebnisse der Mathematik und ihrer Grenzgebiete 9, Springer, Berlin, 1986. · Zbl 0651.53001
[24]	Lalonde, F. and Sikorav, J.-C., Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents, Comment. Math. Helv.66 (1991), 18-33. · Zbl 0759.53022
[25]	Lees, J. A., On the classification of Lagrange immersions, Duke Math. J.43 (1976), 217-224. · Zbl 0329.58006
[26]	Lin, F., Exact Lagrangian caps of Legendrian knots, Preprint, 2013. arXiv:1309.5101. · Zbl 1346.57011
[27]	Murphy, E., Loose Legendrian embeddings in high dimensional contact manifolds, Preprint, 2012. arXiv:1201.2245.
[28]	Murphy, E., Closed exact Lagrangians in the symplectization of contact manifolds, Preprint, 2013. arXiv:1304.6620.
[29]	Ng, L., Computable Legendrian invariants, Topology42 (2003), 55-82. · Zbl 1032.53070
[30]	Polterovich, L., The surgery of Lagrange submanifolds, Geom. Funct. Anal.1 (1991), 198-210. · Zbl 0754.57027
[31]	Sauvaget, D., Curiosités lagrangiennes en dimension 4, Ann. Inst. Fourier (Grenoble)54 (2004), 1997-2020. · Zbl 1071.53046
[32]	Sikorav, J.-C., Some properties of holomorphic curves in almost complex manifolds, No. 117, 165-189 (1994), Basel
[33]	Zehmisch, K., The codisc radius capacity, Electron. Res. Announc. Math. Sci.20 (2013), 77-96. · Zbl 1284.53079