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

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 \((X,d\alpha)\).

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:

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