Symplectic capacities from positive S1–equivariant symplectic homology

Gutt, Jean; Hutchings, Michael

doi:10.2140/agt.2018.18.3537

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Symplectic capacities from positive $S^1$–equivariant symplectic homology

Jean Gutt and Michael Hutchings

Algebraic & Geometric Topology 18 (2018) 3537–3600

DOI: 10.2140/agt.2018.18.3537

Abstract

We use positive $S^{1}$ –equivariant symplectic homology to define a sequence of symplectic capacities $c_{k}$ for star-shaped domains in $ℝ^{2 n}$ . These capacities are conjecturally equal to the Ekeland–Hofer capacities, but they satisfy axioms which allow them to be computed in many more examples. In particular, we give combinatorial formulas for the capacities $c_{k}$ of any “convex toric domain” or “concave toric domain”. As an application, we determine optimal symplectic embeddings of a cube into any convex or concave toric domain. We also extend the capacities $c_{k}$ to functions of Liouville domains which are almost but not quite symplectic capacities.

Keywords

symplectic capacities, equivariant symplectic homology, cube capacity

Mathematical Subject Classification 2010

Primary: 53D05, 53D40, 57R17

References

Publication

Received: 31 October 2017

Revised: 18 May 2018

Accepted: 8 June 2018

Published: 18 October 2018

Authors

Jean Gutt
Mathematical Institute Universität zu Köln Köln Germany

Michael Hutchings
Mathematics Department University of California, Berkeley Berkeley, CA United States

Volume 18, issue 6 (2018)