Gromov’s Compactness Theorem for Pseudo-holomorphic Curves

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Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, unknown): “Gromov compactness for curves with bounded genus and boundary (see, for example, [42, 43]) then implies that the lifted family has a convergent subsequence, which corresponds to a convergent subsequence downstairs.”
Proof of the Arnold chord conjecture in three dimensions I (2011, unknown): “This last fact can also be deduced from standard Gromov compactness theorems – see eg Hummel [3], Wolfson [26], Ye [27] – since the intersection of C with any compact subset of X n† is a pointwise limit of J –holomorphic subvarieties.”
Proof of the Arnold chord conjecture in three dimensions, II (2010, methodology): “ it follows from [26, Lem. 4.4] that C∩(X\Σ) is a J-holomorphic submanifold on the complement of a discrete set. This last fact can also be deduced from standard Gromov compactness theorems, see e.g. [3, 28, 29], since the intersection of Cwith any compact subset of X\ Σ is a pointwise limit of J-ho”
Harmonic maps from degenerating Riemann surfaces (2008, methodology): “An analogue is Gromov’s compactness theorem for pseudo-holomorphic curves [8], [38], [23], [39], [ 12 ].”
$W^{2,2}$-conformal immersions of a closed Riemann surface into $\R^n$ (2010, methodology): “We first collect some facts about degenerating Riemann surfaces from [3, 7].”