The kähler-ricci flow and the ∂¯ operator on vector fields

D. H. Phong*, Jian Song, Jacob Sturm, Ben Weinkove

*Corresponding author for this work

Research output: Contribution to journalArticle

52 Scopus citations

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 ∂¯¯ operator on smooth vector fields is bounded away from 0 along the flow, then the metrics converge exponentially fast in C∞ to a Kähler-Einstein metric.

Original languageEnglish (US)
Pages (from-to)631-647
Number of pages17
JournalJournal of Differential Geometry
Volume81
Issue number3
DOIs
StatePublished - Jan 1 2009

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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