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

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