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Let $M$ be a compact symplectic manifold, $J$ a possibly surface dependent complex structure, and $H$ a Hamiltonian on $M$. I am interested in a variant of Gromov-Floer convergence for solutions of Floer's equation:

$$(du - X_{H} \otimes \gamma) \circ j = J \circ (du - X_{H} \otimes \gamma) $$

for some subclosed one form $\gamma$ on our Riemann surface $(\Sigma,j)$. For simplicity let us suppose that $\Sigma$ is a cylinder with coordinates $f(s)$ and $\gamma=f(s)dt$.

The usual versions of Gromov Floer convergence that I have seen in say McDuff and Salamon's book prove that there is a $C^\infty$ limit (on the smooth part of the curves) given a family of solutions $u_q$ to Floer's equation for pairs $(H_q,J_q)$ which converge in the $C^\infty$ topology to a tuple $(H_\infty,J_\infty)$.

I have also come across this paper:

S. Ivashkovich and V. Shevchishin, Gromov compactness theorem for $J$-complex curves with boundary, Internat. Math. Res. Notices (2000), no. 22, 1167–1206

where they prove $C^0$ convergence to the homogeneous equation where $H=0$ provided we have a family of domain independent complex structures $J_q$ which converge in the $C^0$ topology to a complex structure $J_\infty$ (both in the closed case and the case of Lagrangian boundary). In the closed case, there is also an exposition of this by Tian and Siebert.

http://www.math.uni-hamburg.de/home/siebert/preprints/cime.pdf

Is there a version of this theorem (along with a reference) which allows us to incorporate surface-dependent complex structures and a family of Hamiltonians $H_q$ which converge in the $C^1$ topology to a limit $H_{\infty}$ ? I'd be equally interested in references with partial discussions e.g. that only discuss the case where no bubbles form or there is no breaking along Hamiltonian orbits.

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    $\begingroup$ The straightforward thing to do is to use Gromov's argument that such an inhomogeneous equation is equivalent to the equation for a holomorphic section of some tame almost complex structure that you can write down on the product of the surface with $M$. That of course assumes that the reference you give proves the desired result in the tame case. $\endgroup$ Jun 21, 2014 at 15:19
  • $\begingroup$ Thank you for your suggestion, Mohammed. The first reference only treats compatible complex structures, while the second indeed uses tamed almost complex structures. However, the second only studies the case of closed curves. But, hopefully, between the references and your suggestion I can figure it out. $\endgroup$ Jun 22, 2014 at 3:38

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