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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

We use positive S1–equivariant symplectic homology to define a sequence of symplectic capacities ck for star-shaped domains in 2n. 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 ck 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 ck to functions of Liouville domains which are almost but not quite symplectic capacities.

symplectic capacities, equivariant symplectic homology, cube capacity
Mathematical Subject Classification 2010
Primary: 53D05, 53D40, 57R17
Received: 31 October 2017
Revised: 18 May 2018
Accepted: 8 June 2018
Published: 18 October 2018
Jean Gutt
Mathematical Institute
Universität zu Köln
Michael Hutchings
Mathematics Department
University of California, Berkeley
Berkeley, CA
United States