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We study the problem of convergence of the normalized Ricci flow evolving ona compact manifold $\Omega$ without boundary. In \cite{KS10, KS15} we derived,via PDE techniques, global-in-time existence of the classical solution andpre-compactness of the orbit. In this work we show its convergence tosteady-states, using a gradient inequality of {\L}ojasiewicz type. We have thusan alternative proof of \cite{ha}, but for general manifold $\Omega$ and notonly for unit sphere. As a byproduct of that approach we also derive the rateof convergence according to this steady-sate being either degenerate ornon-degenerate as a critical point of a related energy functional.

Gradient inequality and convergence of the normalized Ricci flow