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In McDuff & Salamon's book "J-holomorphic Curves and Symplectic Topology", only the Gromov compactness for spheres has been proved, and I wonder whether or not this compactness result is still true for genus $g>0$ Riemann surfaces.

If there does exist such a generalization of Gromov compactness, then does there exist any reference that contains detailed proof of it? It seems that the techniques used in this book are hard to apply for the case of arbitrary genus.

I prefer a proof using the notion of stable maps as in this book.

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    $\begingroup$ The first step is to do a very quick google search and/or look for references in the McDuff-Salamon book: "Gromov's compactness theorem for pseudo holomorphic curves" by Rugang Ye (reference 310 of that book). Also see "Pseudo-holomorphic maps and bubble trees" by Parker-Wolfson. $\endgroup$ Nov 11, 2016 at 3:12
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    $\begingroup$ Also see Christoph Hummel's book "Gromov's compactness theorem for pseudo-holomorphic curves" for a detailed geometric proof. $\endgroup$ Oct 25, 2019 at 17:31

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MR1451624 Hummel, Christoph, Gromov's compactness theorem for pseudo-holomorphic curves. Progress in Mathematics, 151. Birkhäuser Verlag, Basel, 1997.

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