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Gromov’s compactness theorem for pseudo-holomorphic curves. (English) Zbl 0810.53024

Let \((M,J)\) be a compact almost complex manifold of real dimension \(2n\), where \(J\) is an almost complex structure on \(M\). Given a Riemann surface \(S\) with complex structure \(j\), a smooth map \(f : (S,j) \to (M,J)\) is called a pseudo holomorphic curve if \(df \circ j = J \circ df\). On a symplectic manifold \(M\), M. Gromov considered in [Invent. Math. 82, 307-347 (1985; Zbl 0592.53025)] all the almost complex structures compatible with the symplectic one and studied the moduli space of pseudo holomorphic curves defined in terms of one such almost complex structure.
The first question one asks about a moduli space is whether it is compact. The total moduli space of pseudo holomorphic curves is very big in general, so one needs to restrict to fixed topological types and impose an area bound. Gromov’s compactness theorem says that a sequence of pseudo holomorphic curves of a fixed topological type and uniformly bounded area converges to a cusp-curve, which is essentially a holomorphic curve joint at isolated points.
In the present interesting paper, the author gives a complete proof of Gromov’s result for pseudo holomorphic curves both in the case of closed curves and curves with boundary.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
35D10 Regularity of generalized solutions of PDE (MSC2000)
58D27 Moduli problems for differential geometric structures

Citations:

Zbl 0592.53025
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