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Let $(M,\lambda)$ be a Liouville manifold. Consider two different contact boundaries $\partial_{\infty}^1M$ and $\partial_{\infty}^2M$ with respect to the same Liouville flow $Z$. Each of them determines a class ${\cal H}_i(M),~i=1,2$ of Hamiltonians which are quadratic at infinity.

Given an admissible Lagrangian $L$ for wrapped Floer theory (i.e. exact, $\lambda|_L\equiv 0$ at infinity and satisfying a non-degeneracy condition for both $\partial_{\infty}^1M$ and $\partial_{\infty}^2M$). We have well-defined wrapped Floer cochain complexes $CW^*(L,L;H_i,J_i),~i=1,2$ for generic Floer data $(H_i,J_i)$ with $H_i\in {\cal H}_i(M)$.

Is there a well-defined continuation map $$CW^*(L,L;H_1,J_1)\rightarrow CW^*(L,L;H_2,J_2)?$$

Improve this question
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  • $\begingroup$ The short answer is yes, at least for contact type almost complex structures. First you rescale to get $H_2 > H_1$ near infinity and interpolate with $H_s$. Then you prove that for $u$ a disk contributing to the continuation map, $H_s\circ u$ satisfies a maximum principle. $\endgroup$
    – Zack
    Commented Oct 18, 2020 at 20:50

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