Hofer geometry of a subset of a symplectic manifold |
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Authors: | Jan Swoboda Fabian Ziltener |
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Institution: | 1. Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany 2. Korea Institute for Advanced Study, 85 Hoegiro (207-43 Cheongnyangni-dong), Dongdaemun-gu, Seoul, 130-722, Korea
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Abstract: | To every closed subset X of a symplectic manifold (M, ω) we associate a natural group of Hamiltonian diffeomorphisms Ham (X, ω). We equip this group with a semi-norm ${\Vert\cdot\Vert^{X, \omega}}$ , generalizing the Hofer norm. We discuss Ham (X, ω) and ${\Vert\cdot\Vert^{X, \omega}}$ if X is a symplectic or isotropic submanifold. The main result involves the relative Hofer diameter of X in M. Its first part states that for the unit sphere in ${\mathbb{R}^{2n}}$ this diameter is bounded below by ${\frac{\pi}{2}}$ , if n ≥ 2. Its second part states that for n ≥ 2 and d ≥ n there exists a compact subset X of the closed unit ball in ${\mathbb{R}^{2n}}$ , such that X has Hausdorff dimension at most d + 1 and relative Hofer diameter bounded below by π / k(n, d), where k(n, d) is an explicitly defined integer. |
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