This paper studies how symplectic invariants created from Hamiltonian
Floer theory change under the perturbations of symplectic structures,
notnecessarily in the same cohomology class. These symplectic invariants include
spectral invariants, boundary depth, and (partial) symplectic quasistates.
This paper can split into two parts. In the first part, we prove some energy
estimations which control the shifts of symplectic action functionals. These
directly imply positive conclusions on the continuity of spectral invariants and
boundary depth in some important cases, including any symplectic surface
and any closed
symplectic manifold
with
.
This follows by applications on some rigidity of the subsets of a symplectic manifold
in terms of heaviness and superheaviness, as well as on the continuity property of
some symplectic capacities. In the second part, we generalize the construction in
the first part to any closed symplectic manifold. In particular, to deal with
the change of Novikov rings from symplectic structure perturbations, we
construct a family of variant Floer chain complexes over a common Novikov-type
ring. In this setup, we define a new family of spectral invariants called
–spectral
invariants, and prove that they are upper semicontinuous under the symplectic
structure perturbations. This implies a quasi-isometric embedding from
to
under
some dynamical assumption, imitating the main result of Usher (Ann. Sci. Éc.
Norm. Supér. 46 (2013) 57–128).
