2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/143854We prove a uniform Sobolev inequality for Ricci flow, which is independent of the number of surgeries. As an application, under less assumptions, a non-collapsing result stronger than Perelman's $κ$ non-collapsing with surgery is derived. The proof is shorter and seems more accessible. The result also improves some earlier ones where the Sobolev inequality depended on the number of surgeries.Differential GeometryAnalysis of PDEs53C21text