In this short notes, we discuss monotonicity formulas under various rescaled versions of Ricci flow. The main result is Theorem 2.1. 1. Functionals Wek from rescaled Ricci flow point of view This is the research notes when the author wrote [Li07]. In the first section, we discuss the relation between functionals Wek(g, f, τ) and rescaled Ricci flow. In Theorem 4.2 [Li07] , we have defined funct...

Abstract. Let M be a compact Riemannian manifold and the metrics g = g(t) evolve by the Ricci flow. We prove the following result. The Sobolev imbedding by Aubin or Hebey, perturbed by a scalar curvature term and modulo sharpness of constants, holds uniformly for (M, g(t)) for all time if the Ricci flow exists for all time; and if the Ricci flow develops a singularity in finite time, then the s...

The study of behavior of the eigenvalues of differential operators along the flow of metrics is very active. We list a few such studies as follows. Perelman [9] proved the monotonicity of the first eigenvalue of the operator −∆ + 1 4 R along the Ricci flow by using his entropy and was then able to rule out nontrivial steady or expanding breathers on compact manifolds. X. Cao [1] and J. F. Li [6...

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 attent...

We consider the Ricci flow for simply connected nilmanifolds, which translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product with respect to time and the evolution of structure constants with respect to time, as well as the evolution of these quantities modulo rescaling. We set up systems of O.D.E.’s for some of these flows and des...

In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is two-fold. One is to give a complete proof of the main theorem of Hamilton in [17]; the other is to extend some results of Perelman [26], [27] to four-manifolds. During the the proof we have actually provided, up to slight modifications...

One of the major tools introduced by Perelman is his reduced volume ̃ V [21, Sect. 7]. This is a certain geometric quantity which is monotonically nondecreasing in time when one has a Ricci flow solution. Perelman’s main use of the reduced volume was to rule out local collapsing in a Ricci flow. Before giving his rigorous proof that ̃ V is monotonic, Perelman gave a heuristic argument [21, Sect. ...

This is an expository article based on the author’s lecture delivered at the conference Lie Theory and Its Applications in March 2011, UCSD. We discuss various notions of positivity and their relations with the study of the Ricci flow, including a proof of the assertion, due to Wolfson and the author, that the Ricci flow preserves the positivity of the complex sectional curvature. We discuss th...

In this paper we study the heat equation (of Hodge Laplacian) deformation of .p; p/-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a .p; p/-form solution is preserved under such an invariant condition, we prove the sharp differential Harnack (in the sense of LiYau-Hamilton) estimates for the positive solutions of the Hodge Laplacian heat equa...

These are notes on Perelman’s papers “The Entropy Formula for the Ricci Flow and its Geometric Applications” [51] and “Ricci Flow with Surgery on Three-Manifolds’ [52]. In these two remarkable preprints, which were posted on the ArXiv in 2002 and 2003, Grisha Perelman announced a proof of the Poincaré Conjecture, and more generally Thurston’s Geometrization Conjecture, using the Ricci flow appr...