Gradient inequality and convergence to steady-states of the normalized Ricci flow on surfaces

In the current work we study the problem of convergence of the normalized Ricci flow evolving on compact Riemannian surfaces without boundary. Indeed in Kavallaris and Suzuki (2010, 2015) global-in-time existence of the classical solution and pre-compactness of the orbit via PDE techniques, are investigated by the authors. The main aim of this work is to show convergence of global-in-time solutions towards steady-states, using a gradient inequality of Łojasiewicz type. Our technique infers an alternative proof of the convergence results presented in Chow (1991) and Hamilton (1988); it also applies to general two-dimensional surfaces and not only to the unit sphere as it happens with the geometric approach developed in Chow (1991) and Hamilton (1988). As a byproduct of our analytical (PDE) approach we obtain the exponential rate of convergence towards a steady-state in case it occurs as a non-degenerate critical point of a related energy functional.