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.
This theorem, and its generalizations to punctured pseudoholomorphic curves, underlies the compactness results for flow lines in Floer homology and symplectic field theory.
Usually, the energy bound is achieved by considering a symplectic manifold with compatible almost-complex structure as the target, and assuming that curves to lie in a fixed homology class in the target.
The finiteness of the bubble tree follows from (positive) lower bounds on the energy contributed by a holomorphic sphere.