Perelman's W-functional and stability of Kähler-Ricci flow

Abstract

In this expository note, we study the second variation of Perelman's entropy on the space of Kahler metrics at a Kähler-Ricci soliton. We prove that the entropy is stable in the sense of variations. In particular, Perelman's entropy is stable along the Kähler-Ricci flow. The Chinese version of this note has appeared in a volume in honor of professor K.C.Chang (Scientia Sinica Math., 46 (2016), 685-696).

References in corpus (6)

• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Ricci flow on Kähler-Einstein manifolds
• Gaussian densities and stability for some Ricci solitons
• Space of Ricci flows (II)
• Convergence of a Kähler-Ricci flow
• On the Stability of Kähler-Einstein Metrics

Cited by in corpus (12)

• Bessel Functions, Heat Kernel and the Conical Kähler-Ricci Flow
• Supremum of Perelman's entropy and Kähler-Ricci flow on a Fano manifold
• Convergence of Kähler-Ricci flow on Fano manifolds, II
• Ricci flow on Orbifold
• Stability of Kähler-Ricci flow
• Kähler Ricci flow on Fano manifolds with vanished Futaki invariants
• Lagrangian F-stability of closed Lagrangian self-shrinkers
• Stability of Kähler-Ricci flow in the space of Kähler metrics
• The total second variation of Perelman's $\mathcal{W}$-functional
• Kähler Ricci flow with vanished Futaki invariant
• Kähler-Ricci flow for deformed complex structures
• A second variation formula for Perelman's $\mathcal{W}$-functional along the modified Kähler-Ricci flow