Gromov's compactness theorem (topology)

For Gromov's compactness theorem in Riemannian geometry, see that article.

In the mathematical field of symplectic topology, Gromov's compactness theorem states that a sequence of pseudoholomorphic curves in an almost complex manifold with a uniform energy bound must have a subsequence which limits to a pseudoholomorphic curve which may have nodes or (a finite tree of) "bubbles". A bubble is a holomorphic sphere which has a transverse intersection with the rest of the curve. If the complex structures on the curves in the sequence do not vary, only bubbles may occur (equivalently, the curves that pinch to cause the degeneration of the limiting curve must be contractible). If the complex structures is allowed to vary, nodes can occur as well. Usually, the area bound is achieved by considering a symplectic manifold with compatible almost-complex structure as the target and restricting the images of the curves to lie in a fixed homology class. This theorem underlies the compactness results for flow lines in Floer homology.

References

• M. Gromov, Pseudo holomorphic curves in symplectic manifolds. Inventiones Mathematicae, vol. 82, 1985, pp. 307-347.