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Quantitative h-principle in symplectic geometry

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Abstract

We prove a quantitative h-principle statement for subcritical isotropic embeddings. As an application, we construct a symplectic homeomorphism that takes a symplectic disc into an isotropic one in dimension at least 6.

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Notes

  1. Recall that this means there exists a continuous map \(G:[0,1]^2\times \text {Op}\,(B)\rightarrow G^{\text {iso}}(k,n)\) such that \(G(0,t,z)=G_t(z)\) and \(G(1,t,z)=\mathrm{d}v_t(z)\) \(\forall (t,z)\in [0,1]\times \text {Op}\,(B)\), \(G(s,t,z)=\mathrm{d}u_0(z)\) \(\forall (s,t,z)\in [0,1]^2\times \text {Op}\,(A\cap B)\), \(G(s,0,z)=G_0(z)=\mathrm{d}u_0(z)\) and \(G(s,1,z)=G_1(z)=\mathrm{d}v_1(z)\) \(\forall (s,z)\in [0,1]\times \text {Op}\,(B)\).

References

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Acknowledgements

We are very much indebted to Claude Viterbo for his support and interest in our research. Claude’s fundamental contributions to symplectic geometry and topology, and in particular to the field \(\mathbb {\mathcal {C}}^0 \) symplectic geometry, are widely recognized. We wish Claude all the best, and to continue enjoying math and delighting us with his creative and inspiring mathematical works.

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Correspondence to Lev Buhovsky.

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Dedicated to Claude Viterbo, on the occasion of his 60th birthday. This article is part of the topical collection “Symplectic geometry-A Festschrift in honour of Claude Viterbo’s 60th birthday” edited by Helmut Hofer, Alberto Abbondandolo, Urs Frauenfelder, and Felix Schlenk.

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Buhovsky, L., Opshtein, E. Quantitative h-principle in symplectic geometry. J. Fixed Point Theory Appl. 24, 38 (2022). https://doi.org/10.1007/s11784-022-00947-8

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  • DOI: https://doi.org/10.1007/s11784-022-00947-8

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