2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/108983We give a proof to the Li-Yau-Hamilton type inequality claimed by Perelman on the fundamental solution to the conjugate heat equation. The rest of the paper is devoted to improving the known differential inequalities of Li-Yau-Hamilton type via monotonicity formulae.Differential Geometry58J35text