Gromov's compactness theorem (topology)
Encyclopedia
In 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
.
Symplectic topology
Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form...
, Gromov's compactness theorem states that a sequence of pseudoholomorphic
Pseudoholomorphic curve
In mathematics, specifically in topology and geometry, a pseudoholomorphic curve is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann equation. Introduced in 1985 by Mikhail Gromov, pseudoholomorphic curves have since revolutionized the study of...
curves in an almost complex manifold
Almost complex manifold
In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex manifold. That is, every complex manifold is an almost...
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
Floer homology
Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite...
.