Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries after deforming one of the complex structures. In other words, when does there exist $(X_2,J_2')$ (the same underlying smooth manifold with a perhaps different complex structure) with a biholomorphism on a neighborhood of the boundary between $(X_1,J_1)$ and $(X_2,J_2')$. Are there obstructions to this? I am primarily interested in the case with $\Bbb C$ dimension equal to two.
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1$\begingroup$ I'm out of my depth here, but isn't the answer given by the solution to the CR equivalence problem? See e.g. projecteuclid.org/download/pdf_1/euclid.bams/1183548684 $\endgroup$– Vít TučekCommented Apr 13, 2016 at 7:44
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$\begingroup$ @VítTuček Thanks for the helpful comment. There's a theorem of Eliashberg that the map CR-str. to contact str. is a homotopy equivalence but I don't know what kind of setting that is exactly in. If this is in the right setting, I guess I need to see if these deformations can be lifted to deformations of the complex structure in the neighborhood. $\endgroup$– PVALCommented Apr 18, 2016 at 22:37
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