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Gromov's compactness theorem for pseudo-holomorphic curves, in section 1.5 of "Pseudoholomorphic curves in symplectic manifolds," is very well known. I'm aware of the following two proofs:

  • J.G. Wolfson. Gromov's compactness of pseudo-holomorphic curves and symplectic geometry. J. Differential Geom. Volume 28, Number 3 (1988), 383-405.
  • Rugang Ye. Gromov's compactness theorem for pseudo holomorphic curves. Trans. Amer. Math. Soc. 342 (1994), 671-694.

Wolfson's proof applies the methods of

  • J. Sacks and K. Uhlenbeck. The existence of minimal immersions of 2-spheres. Ann. of Math. (2) 113 (1981), no. 1, 1–24.
  • Richard Schoen and Karen Uhlenbeck. A regularity theory for harmonic maps. J. Differential Geometry 17 (1982), no. 2, 307–335.

which seems quite natural as Gromov's theorem is an adaptation of the Sacks-Uhlenbeck phenomena for harmonic maps to pseudoholomorphic curves. Ye explains that Gromov's proof in the case of manifolds with boundary (section 1.5D2 of his paper) does not work in general. He also explains that Wolfson's result is weaker than Gromov's, since it does not recover area convergence; and that Wolfson's methods do not immediately extend to the case of manifolds with boundary. Ye gives a proof of Gromov's theorem for manifolds with and without boundary, and with area convergence. (To my knowledge it is the first to do so, if Gromov's only works for closed manifolds.)

My questions are:

  • Is it possible to extend Wolfson's methods to manifolds with boundary, and has anyone done so? I am thinking of the following article as inspiration:

    Richard Schoen and Karen Uhlenbeck. Boundary regularity and the Dirichlet problem for harmonic maps. J. Differential Geom. 18 (1983), no. 2, 253–268.
  • Is it possible to extend Wolfson's methods to recover area convergence, and has anyone done so? The issue seems conceptually parallel to the problem of energy quantization for harmonic maps, which has been widely studied. The following article, for instance, applies the Sacks-Uhlenbeck theory to get such a result.

    Yum Tong Siu and Shing Tung Yau. Compact Kähler manifolds of positive bisectional curvature. Invent. Math. 59 (1980), no. 2, 189–204.
  • Are there other proofs (with details) that give Ye's result?
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    $\begingroup$ It's not exactly what you're looking for, but Joel Fish's "Target-local Gromov compactness theorem" (arxiv.org/abs/0912.4435) gives a Gromov compactness theorem for $J$-holomorphic curves with boundary that live inside a manifold with boundary. It relies on the interpretation of $J$-holomorphic curves as surfaces satisfying an inhomogeneous mean curvature equation. $\endgroup$
    – Rohil Prasad
    Commented Sep 9, 2020 at 0:37

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