Gradient inequality and convergence to steady-states of the normalized Ricci flow on surfaces. (August 2022) - Kavallaris, Nikos I.; Suzuki, Takashi
Abstract: 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.