Gradient inequality and convergence of the normalized Ricci flow

We study the problem of convergence of the normalized Ricci flow evolving on a compact manifold Ω without boundary. In [11, 12] we derived, via PDE techniques, global-intime 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 Lojasiewicz type. We have thus an alternative proof of [7], but for general manifold Ω 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.

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