Gromov's compactness theorem (geometry)

Gromov's compactness theorem, often referred to in the context of many areas in geometric analysis and differential geometry, primarily deals with the compactness of certain collections of Riemannian manifolds. It provides a criterion for when a sequence of Riemannian manifolds can be shown to converge in a meaningful way. The theorem applies to families of Riemannian manifolds that are uniformly bounded in terms of geometry, meaning they satisfy certain bounds on curvature, diameter, and volume.