Gradient inequality and convergence to steady-states of the normalized Ricci flow on surfaces. (August 2022)

• Record Type: Journal Article
• Authors: Kavallaris, Nikos I.; Suzuki, Takashi
• Is Part Of: Nonlinear analysis. Volume 221(2022)
• Publication Date: 2022-08
• Subjects: primary 35K55 53C44 -- secondary 35B40; Ricci flow -- Gradient inequality -- Logarithmic diffusion; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112906
• Languages: English
• ISSNs: 0362-546X

Abstract

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.