## Symplectic embeddings and the Lagrangian bidisk.(English)Zbl 1370.53057

In the study of symplectic manifolds symplectic embedding problems play a significant role. Many techniques were created to deal with questions about when symplectic embeddings exist. Symplectic capacities are one such technique, and they provide an obstruction to the existence of a symplectic embedding. If for a symplectic four-dimensional manifold $$(X,\omega)$$, there is a sequence of real numbers $$c_k(X,\omega)$$ such that $0=c_0(X,\omega)<c_1(X,\omega)\leq c_2(X,\omega)\leq\dots\leq\infty,$ then they are called embedded contact homology (ECH) capacities. The capacities satisfy the following properties:
(i) if $$a>0$$, then $$c_k(X,a\cdot\omega)=a\cdot c_k(X,\omega)$$ for every $$k$$,
(ii) if $$(X_1,\omega_1)\hookrightarrow(X_2,\omega_2)$$, then $$c_k(X_1,\omega_1)\leq c_k(X_2,\omega_2)$$ for all $$k$$, and
(iii) $$c_k\left(\coprod\limits_{i=1}^n(X_i,\omega_i)\right)=\max\left\{\sum\limits_{i=1}^nc_{k_i}(X_i,\omega_i);\;k_1+\dots+k_n=k\right\}$$.
For a certain embedding problem $$(X_1,\omega_1)\hookrightarrow(X_2,\omega_2)$$ ECH capacities are said to be sharp if $$\forall_{k}\;c_k(X_1,\omega_1)\leq c_k(X_2,\omega_2)\;\Longrightarrow\;(X_1,\omega_1)\hookrightarrow(X_2,\omega_2)$$. Any subset of $$\mathbb R^4\cong\mathbb C^2$$ with coordinates $$(p_1,q_1,p_2,q_2)=(z_1,z_2)$$ is endowed with the symplectic form $$\omega=\sum\limits_{i=1}^2dp_i\wedge dq_i$$. A Lagrangian bidisk $$P_L$$ in $$\mathbb R^4$$ is defined as $P_L=\{(p_1,q_1,p_2,q_2);\;p_1^2+p_2^2\leq1 \wedge q_1^2+q_2^2\leq1\}$ and the Lagrangian product of any two disks is symplectomorphic to a multiple of $$P_L$$. An ellipsoid $$E(a,b)$$ and a symplectic polydisk $$P(a,b)$$ are defined as $\begin{gathered} E(a,b)=\left\{(z_1,z_2);\;\pi\left(\frac{|z_1|^2}a+\frac{|z_2|^2}b\right)\leq 1\right\}\\ P(a,b)=\{(z_1,z_2);\;\pi|z_1|^2\leq a \wedge \pi|z_2|^2\leq b\}, \end{gathered}$ respectively. $$B(a)=E(a,a)$$ is the Euclidean ball of radius $$\sqrt{a/\pi}$$.
In this paper, the author obtains sharp obstructions to the symplectic embedding of the Lagrangian bidisk into four-dimensional balls, ellipsoids, and symplectic polydisks.
It is proven that ECH capacities give a sharp obstruction to symplectically embedding the interior of $$P_L$$ into balls, ellipsoids, and symplectic polydisks. Moreover,
(i) $$\text{int}(P_L)\hookrightarrow B(a)$$ if and only if $$a\geq 3\sqrt 3$$,
(ii) $$\text{int}(P_L)\hookrightarrow E(a,b)$$ if and only if $$\min(a,b)\geq 4$$ and $$\max(a,b)\geq 3\sqrt3$$, and
(iii) $$\text{int}(P_L)\hookrightarrow P(a,b)$$ if and only if $$a,b\geq 4$$.
If $$\Omega$$ is a closed region in the first quadrant of $$\mathbb R^2$$, then the toric domain $$X_\Omega\subset\mathbb C^2$$ is defined as $$X_\Omega=\{(z_1,z_2);\;\pi(|z_1|^2,|z_2|^2)\in\Omega\}$$ and is endowed with the restriction of the standard symplectic form in $$\mathbb C^2$$. To prove the main result the author needs to show that if $$X_0$$ is the toric domain $$X_{\Omega_0}$$, where $$\Omega_0$$ is the region bounded by the coordinate axes and the curve parameterized by $\left(2\sin\left(\frac{\alpha}2\right)-\alpha\cos\left(\frac{\alpha}2\right),2\sin\left(\frac{\alpha}2\right)+(2\pi-\alpha)\cos\left(\frac{\alpha}2\right)\right),\;\;\alpha\in[0,2\pi],$ then $$\text{int}(P_L)$$ and $$\text{int}(X_0)$$ are symplectomorphic.

### MSC:

 53D05 Symplectic manifolds (general theory) 53D42 Symplectic field theory; contact homology
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