The equivalence of Ekeland-Hofer and equivariant symplectic homology capacities

TL;DR

This paper proves the equivalence of Ekeland-Hofer and equivariant symplectic homology capacities on star-shaped domains, resolving a 35-year-old open question in symplectic geometry.

Contribution

It establishes the equality of two important symplectic capacities and links Floer homology with Morse homology for Hamiltonians.

Findings

Ekeland-Hofer capacities and equivariant symplectic homology capacities are equal on star-shaped domains.

Floer homology of Hamiltonians is chain-complex isomorphic to Morse homology of the action functional.

Resolved a 35-year-old open problem in symplectic capacity theory.

Abstract

In this paper, we prove that the Ekeland-Hofer capacities coincide on all star-shaped domain in $\mathbb{R}^{2n}$ with the equivariant symplectic homology capacities defined by the first author and Hutchings, answering a 35 years old question. Along the way, we prove that given a Hamiltonian $H$, the (equivariant) Floer homology of $H$ is chain-complex isomorphic to the Morse homology of the Hamiltonian action functional.