Symplectic structure perturbations and continuity of symplectic invariants

Abstract

This paper studies how some symplectic invariants which are born from Hamiltonian Floer theory (e.g. spectral invariant, boundary depth, (partial) symplectic quasi-state) change with respect to symplectic structure perturbations, i.e., new symplectic structures perturbed from a known symplectic structure. This paper can be roughly divided into two parts. In the first part, we will prove a family of energy estimation inequalities which control the shifts of action functional in the Hamiltonian Floer theory. This directly implies an affirmative conclusion on continuity of spectral invariant and boundary depth in several important cases, for instance, the symplectic surface Σ g>1 or closed symplectic manifold M with dim H 2 (M; ) = 1. This follows by an application on the rigidity of subsets on symplectic manifolds in terms of heavy or superheavy. In the second part, we generalize the construction in the first part to any symplectic manifold. Specifically, in order to deal with the change of Novikov ring due to the perturbations, we will construct a (local) family of variant Floer chain complexes over a variant Novikov ring and study its homologies, which takes its inspiration from Ono's construction in [Ono05]. We will also prove, in this set-up, a new family of spectral invariant called t-spectral invariant is upper semicontinuous. This has applications on a quasi-embedding from ∞ to H am(M, ω) under a certain dynamical condition imitating the main result from [Ush13] and several continuity properties of Hofer-Zehnder capacity and spectral capacity. Finally, t-boundary depth (defined over the local family of variant Floer chain complexes above) is defined and briefly discussed but its continuity property is unknown.

Invariance of symplectic cohomology and twisted cotangent bundles over surfaces

“…In retrospect (unknown to the author at the time) the idea involved is similar to estimates in Le-Ono [37,Lemma 5.4]: they do not deform ω, but [37, Theorem 5.3] builds a continuation map arising from a deformation of the symplectic vector field, similar to the one we construct in Section 5. This energy estimate has since appeared independently in the work of Zhang on spectral invariants for aspherical closed symplectic manifolds [60,Sec.4]. The key idea of the energy estimate is also used at the heart of the recent work of Groman-Merry on magnetic geodesics [29, Theorem 6.2] (compare with Theorem 4.8).…”

Disjoint superheavy subsets and fragmentation norms

“…We use the following result to prove Corollary 2.6. Proposition 7.4 ([Pol12], [Ka17], [Ish,Proposition 4.4], [Zha,Theorem 1.9]). For any positive integer g, there exists a positive number K such that c (Σg ;Z2) ( φF ) + F ≤ K…”

The symplectic cohomology of magnetic cotangent bundles

“…the convex hull of finitely many points). We now introduce a generalisation of the Novikov ring, which we denote by Λ K Φ (Γ), which also appeared in recent work of Zhang [46]. In the case Φ = {φ}, this reduces to the previously defined Novikov ring Λ K φ (Γ).…”

“…This would be slightly more natural in light of the fact that our symplectic cohomology groups SH * (T * N; ω σ ) depend on σ only through the half-line {r 11). Nevertheless, we stick with polytopes to be consistent with [46].…”

“…This means we are working with a strictly non-quantitative version of symplectic cohomology. The behavior of quantitative Floer-theoretic invariants under deformation of the symplectic structure is less wellbehaved and has been studied in [46]. Remark 1.10.…”