Perelman’s W-functional and Stability of Kähler-ricci Flow

The Sasaki-ricci Flow and Compact Sasaki Manifolds of Positive Transverse Holomorphic Bisectional Curvature

We show that Perelman’s W functional on Kahler manifolds has a natural counterpart on Sasaki manifolds. We prove, using this functional, that Perelman’s results on Kahler-Ricci flow (the first Chern class is positive) can be generalized to Sasaki-Ricci flow, including the uniform bound on the diameter and the scalar curvature along the flow. We also show that positivity of transverse bisectiona...

On the lower bound of energy functional E1 (I) —a stability theorem on the Kähler Ricci flow

Eigenvalues and energy functionals with monotonicity formulae under Ricci flow

Abstract. In this note, we construct families of functionals of the type of F-functional and W-functional of Perelman. We prove that these new functionals are nondecreasing under the Ricci flow. As applications, we give a proof of the theorem that compact steady Ricci breathers must be Ricci-flat. Using these new functionals, we also give a new proof of Perelman’s no non-trivial expanding breat...

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 attent...

Non-kähler Ricci Flow Singularities That Converge to Kähler–ricci Solitons

We investigate Riemannian (non-Kähler) Ricci flow solutions that develop finite-time Type-I singularities with the property that parabolic rescalings at the singularities converge to singularity models taking the form of shrinking Kähler–Ricci solitons. More specifically, the singularity models for these solutions are given by the “blowdown soliton” discovered in [FIK03]. Our results support th...