Symplectic isotopies in dimension greater than four
The author shows that certain spaces of symplectic embeddings of a polydisk into \(B^4 \times \mathbb{R}^{2(n-2)}\) are not path connected for \(2n \geq 6\), and any pair of such nonisotopic embeddings do not extend to the same ellipsoid. Consider \(\mathbb{R}^{2n}\) with its standard symplectic structure given by coordinates \(x_j\), \(y_j\) for \(1 \leq j \leq n\), and let \(z_j = x_j + i y_j\). The ellipsoid \[ E(a_1,\ldots, a_n) = \left\{\sum_j \frac{\pi |z_j|^2}{a_j} \leq 1 \right\}, \] the polydisk \[ P(a_1,\ldots, a_n) = \left\{\pi |z_j|^2 \leq a_j \text{ for all } j \right\}, \] and the polylike domain \[ Q(b,a_2,\ldots,a_n) = \left\{\pi |z_1|^2 \leq b, \sum_{j=2}^n \frac{\pi |z_j|^2}{a_j} \leq 1 \right\}, \] are symplectic submanifold of \({\mathbb R}^{2n}\). Generalizing a result in [Algebr. Geom. Topol. 13, No. 4, 2171--2192 (2013; Zbl 1292.53054)] the author proves the following. Theorem 1.3. Suppose that \(a_2 < b\) and \(a_j > 2a_2\) for all \(j \geq 3\). There does not exist a Hamiltonian diffeomorphism \(\phi\) with support contained in \(\overset{\circ}{B^4}(2a_2 + b) \times \mathbb{R}^{2(n-2)}\) such that \(\phi(Q(b,a_2,\ldots, a_n)) \subset \overset{\circ}{B^4}(2a_2 + b) \times \mathbb{R}^{2(n-2)}\). Corollary 1.4. Let \(a_3, \ldots, a_n > 2a_2\) and choose \(R\) with \(a_2 + b < R < 2a_2 + b\). Suppose either \(2a_2 < b\) or \(a_3 < a_2 + b\). Then there exists a symplectic embedding \(Q(b,a_2,\ldots ,a_n) \rightarrow B^4(R) \times \mathbb{R}^{2(n-2)}\) which is not symplectically isotopic to the inclusion inside \(B^4(R) \times \mathbb{R}^{2(n-2)}\). Previous results of the author show that the preceding bound on \(R\) is sharp. Theorem 1.3 gives results about embeddings of polydisks that are not isotopic to the inclusion, and the author also proves the following directly, which in some cases generalizes a result from [\textit{A. Floer} et al., Math. Z. 217, No. 4, 577--606 (1994; Zbl 0869.58012)]. Theorem 1.5. Let \(a_1 \leq \cdots \leq a_n\) with \(a_3 > \max(2a_1,a_2)\) and \(a_1 + a_3 < R < 2a_1 + a_3\). Then the space of embeddings \(P(a_1,\ldots , a_n) \rightarrow B^4(R) \times \mathbb{R}^{2(n-2)}\) is not path connected. More precisely, the embedding \(f:(z_1,z_2, z_3, \ldots, z_n) \mapsto (z_1, z_3, z_2, z_4, \ldots , z_n)\) is not isotopic to any map with image contained in \(\overset{\circ}{B^4}(a_1 + a_3) \times \mathbb{R}^{2(n-2)}\). In particular, the inclusion is not isotopic to \(f\). Generalizing a result from [\textit{D. McDuff}, J. Topol. 2, No. 1, 1--22 (2009; Zbl 1166.53051)] the author also proves Theorem 1.9. i. Let \(Q = Q(b,a_2, \ldots, a_n)\) be a polylike domain with \(a_2 < b\) and \(a_j > 2a_2\) for all \(j \geq 3\), and let \(E = E(c_1, \ldots , c_n)\) be an ellipsoid with \(Q \subset E\). Let \(R < 2a_2 + b\). There do not exist embeddings \(f_0, f_1: E \rightarrow B^4(R) \times \mathbb{R}^{2(n-2)}\) such that \(f_0\) restricts to the boundary on \(Q\) and \(f_1(Q) \subset \overset{\circ}{B^4}(a_2 + b) \times \mathbb{R}^{2(n-2)}\) ii. Let \(P = P(a_1, \ldots , a_n)\) be a polydisk with \(a_1 \leq \cdots \leq a_n\) and \(a_3 > \max(2a_1,a_2)\), and let \(R\) satisfy \(a_1 + a_3 < R < 2a_1 + a_3\). Let \(E = E(c_1,\ldots ,c_n)\) be an ellipsoid with \(P \subset E\). There do not exist embeddings \(f_0,f_1:E \rightarrow B^4(R) \times \mathbb{R}^{2(n-2)}\) such that \(f_0|_P\) is given by \((z_1,z_2,z_3, \ldots ,z_n) \mapsto (z_1,z_3,z_2,z_4, \ldots ,z_n)\) and \(f_1(P) \subset \overset{\circ}{B^4}(a_1 + a_3) \times \mathbb{R}^{2(n-2)}\). Section 2 of the paper contains the proof of Theorem 1.3, Section 3 contains the proof of Theorem 1.5, and Section 4 contains the proof of Theorem 1.9. Section 4 also contains more nonextension results that follow from the other theorems in the paper.
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