2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/213003In this article we get a time-dependent Sobolev inequality along the Ricci flow which generalizes the earlier results of Zhang, Ye, Hsu. As an application of the time-dependent Sobolev inequality, we also get a growth of the ratio of bob-collapsing along the Ricci flow.Proof of Theorem 2.2 is added, a serious mistake in the proof of Theorem 2.3 is correctedDifferential GeometryAnalysis of PDEstext