Maximum principles in symplectic homology

Abstract

In the setting of symplectic manifolds which are convex at infinity, we use a version of the Aleksandrov maximum principle to derive uniform estimates for Floer solutions that are valid for a wider class of Hamiltonians and almost complex structures than is usually considered. This allows us to extend the class of Hamiltonians which one can use in the direct limit when constructing symplectic homology. As an application, we detect elements of infinite order in the symplectic mapping class group of a Liouville domain and prove existence results for translated points.

Hamiltonian perturbations in contact Floer homology

“…Moreover, we have lim [21]). This implies that the canonical map HF * (ε) → SH * (W ) is an isomorphism.…”

“…Our examples come from the two extreme situations where either SH * (W ) = 0 or SH * (W ) is infinite dimensional. In the latter case, even a stronger conclusion than that of Theorem 1.4 holds: there are no contractible positive loops of contactomorphisms on ∂W if SH * (W ) is infinite dimensional (see Theorem 1.5 in [21]).…”

“…Mimicking the study of the Hamiltonian dynamics in symplectic geometry, one way to analyze the dynamical data from (1.1) is to associate a Floer-type homology theory to (M, ξ = ker α). Based on the generalized maximum principle in [21], one can construct such a homology theory, called the contact Floer homology of h and denoted by HF * (W, h) or simply HF * (h) (if it is clear from the context what the filling W is). If the flow φ t has no 1-periodic orbits, HF * (h) is well-defined as the Hamiltonian Floer homology HF * (H) for a Hamiltonian H : [0, 1] × W → R on the completion W of the Liouville domain W .…”

“…This implies that the canonical map HF * (ε) → SH * (W ) is an isomorphism. By the last equality in [21], the contact Floer homology HF * (ε) is isomorphic to H * +n (W, ∂W ). Therefore, we obtain the desired contradiction and this finishes the proof.…”

“…

We study the contact Floer homology HF * (W, h) introduced by Merry-Uljarević in [21], which associates a Floer-type homology theory to a Liouville domain W and a contact Hamiltonian h on its boundary. The main results investigate the behavior of HF * (W, h) under the perturbations of the input contact Hamiltonian h. In particular, we provide sufficient conditions that guarantee HF * (W, h) to be invariant under the perturbations.

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