  We study the problem of convergence of the normalized Ricci flow evolving on
a compact manifold $\Omega$ without boundary. In \cite{KS10, KS15} we derived,
via PDE techniques, global-in-time existence of the classical solution and
pre-compactness of the orbit. In this work we show its convergence to
steady-states, using a gradient inequality of {\L}ojasiewicz type. We have thus
an alternative proof of \cite{ha}, but for general manifold $\Omega$ and not
only for unit sphere. As a byproduct of that approach we also derive the rate
of convergence according to this steady-sate being either degenerate or
non-degenerate as a critical point of a related energy functional.
