2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/141429In this paper, we define a reduced distance function based at a point at the singular time $T<\infty$ of a Ricci flow. We also show the monotonicity of the corresponding reduced volume based at time T, with equality iff the Ricci flow is a gradient shrinking soliton. Our curvature bound assumption is more general than the type I condition.17 pagesDifferential GeometryAnalysis of PDEs58J35; 35K05text