Symplectic isotopies in dimension greater than four
type: Item
instance of: scholarly article
title: Symplectic isotopies in dimension greater than four
zbMATH Open document ID: 1366.53066
published in: The Journal of Symplectic Geometry
publication date: 2017-02-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1403.3846
review text: 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 \(a_1 \leq \cdots \leq a_n\) w…
Mathematics Subject Classification ID: 53D35
Mathematics Subject Classification ID: 53D42
Mathematics Subject Classification ID: 57R17
zbMATH DE Number: 6682855
zbMATH Keywords: symplectic embeddings
zbMATH Keywords: symplectically isotopic
zbMATH Keywords: ellipsoid
zbMATH Keywords: polydisk
author: R. Hind
reviewed by: David E. Hurtubise
MaRDI profile type: Publication
arXiv ID: 1403.3846
OpenAlex ID: W2568580536
DOI: 10.4310/JSG.2016.V14.N4.A3
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description: scientific article