Gromov compactness theorem

Gromov's compactness theorem is a foundational result in geometry, particularly in the study of shapes called "holomorphic curves" within certain spaces. It states that if you have a sequence of such curves with bounded energy (or size), then, after potentially choosing a subsequence, these curves will converge to a limit curve—possibly with some "bubbles" or singularities forming. This ensures the space of these curves is "compact," meaning they can be neatly-studied and understood, even as they deform or vary. It provides a critical tool for analyzing the geometry and topology of complex spaces.