The Kähler-Ricci flow and the $\bar{\partial }$ operator on vector fields

Abstract

The limiting behavior of the normalized Kähler-Ricci flow for manifolds with positive first Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi K-energy is bounded from below, then the scalar curvature converges uniformly to a constant. Second, it is shown that if the Mabuchi K-energy is bounded from below and if the lowest positive eigenvalue of the ${\bar{\partial }}^{†}\bar{\partial }$ operator on smooth vector fields is bounded away from 0 along the flow, then the metrics converge exponentially fast in $C^{\infty }$ to a Kähler-Einstein metric.

16 pages. Final version, to appear in J. Differential Geometry

References in corpus (2)

• Ricci flow on Kähler-Einstein manifolds
• Multiplier Ideal Sheaves and the Kähler-Ricci Flow

Cited by in corpus (3)

• Stability of Kähler-Ricci flow
• A note on compact Kähler-Ricci flow with positive bisectional curvature
• The Kahler-Ricci flow and K-stability