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Nonsqueezing property of contact balls. (English) Zbl 1372.53087

If \(V\) is a contact manifold containing two open domains \(U_1\), \(U_2\), then \(U_1\) can be squeezed into \(U_2\), if there exists a compactly supported contact isotopy \(\Phi_s : \bar{U_1}\longrightarrow V, s \in [0,1]\), such that \(\Phi_0 = \mathrm{Id}\) and \(\Phi_1(\bar{U_1})\subset U_2\), see [Y. Eliashberg et al., Geom. Topol. 10, 1635–1748 (2006; Zbl 1134.53044)]. Let \(B_R\) be the open ball of radius \(R\) in \(\mathbb{R}^{2n}\) and let \(\mathbb{R}^{2n}\times \mathbb{S}^{1}\) be the prequantization space equipped with the standard contact structure. The remaining open case in the above cited paper (namely a contact nonsqueezing conjecture) is covered by the main result of the present paper, which states that if \(R\) and \(r\) satisfy \(1\leq \pi r^2 < \pi R^2\), then it is impossible to squeeze the contact ball \(B_R \times \mathbb{S}^1\) into \(B_r \times \mathbb{S}^1\) via compactly supported contact isotopies. This paper uses terminology and results from the theory of algebraic microlocal analysis. The major machinery relied on is about sheaves and their microlocal singular supports which are developed in [M. Kashiwara and P. Schapira, Sheaves on manifolds. With a short history “Les débuts de la théorie des faisceaux” by Christian Houzel. Berlin etc.: Springer-Verlag (1990; Zbl 0709.18001)]. After dealing with the contact manifold \(\mathbb{R}^{2n}\times \mathbb{S}^1\), the construction of the projector, and the contact isotopy invariants, the present author uses Tamarkin’s idea to prove the main result here with microlocal category methods.

MSC:

53D35 Global theory of symplectic and contact manifolds
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