2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/154107We show that there exists a suitable neighborhood of a constant curvature hyperbolic metric such that, for all initial data in this neighborhood, the corresponding solution to a normalized cross curvature flow exists for all time and converges to a hyperbolic metric. We show that the same technique proves an analogous result for Ricci flow. Additionally, we show short time existence and uniqueness of cross curvature flow for a more general class of initial data than was previously known.Revised version including additional references and enhanced exposition. To appear in PAMSDifferential Geometry53C44; 53C21; 35K55text