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.


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