2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/122658In this note, we construct families of functionals of the type of $\mathcal{F}$-functional and $\mathcal{W}$-functional of Perelman. We prove that these new functionals are nondecreasing under the Ricci flow. As applications, we give a proof of the theorem that compact steady Ricci breathers must be Ricci-flat. Using these new functionals, we also give a new proof of Perelman's no non-trivial expanding breather theorem. Furthermore, we prove that compact expanding Ricci breathers must be Einstein by a direct method. In this note, we also extend X. Cao's methods of eigenvalues\cite{C} and improve their results.19 pages, one reference added, to appear in Mathematische AnnalenDifferential GeometryAnalysis of PDEs53C44; 35K55text