On Perelman’s functional with curvature corrections

In recent ten years, there has been much concentration and increased research activities on Hamilton’s Ricci flow evolving on a Riemannian metric and Perelman’s functional. In this paper, we extend Perelman’s functional approach to include logarithmic curvature corrections induced by quantum effects. Many interesting consequences are revealed. During the last decades, there has been more attention focused on the Ricci flow which was introduced in 1982 by Hamilton [19, 20, 21] and extended later by Perelman [28, 26, 27]. In fact, the Ricci flow is a system of the 2nd order nonlinear weakly parabolic partial differential equations (PDEs) on the metric which can be viewed as a nonlinear heat equation of a metric and contains at most the second derivative of the metric which led Perelman to the proof of the famous Thurston’s geometrization conjecture [29]. In fact, Ricci flow plays a crucial role in string theory [5] as it describes the flow energy effective action. Besides, the Ricci flow plays a significant role in the thermodynamics of black holes [22] which is one of the most important objects in string theory. Strings, higher order curvature and black holes have been found to be inextricably knotted [25]. In string theory, there are higher curvature corrections in addition to the Einstein–Hilbert term. Different forms of curvature corrections may be added [6], nevertheless, a logarithmic correction of the form may be induced by quantum effects and 2000 Mathematics Subject Classification. 49S05, 58A05.