On the convergence and singularities of the J-flow with applications to the Mabuchi energy

Jian Song, Ben Weinkove

Research output: Contribution to journalArticle

51 Citations (Scopus)

Abstract

The J-flow of S. K. Donaldson and X. X. Chen is a parabolic flow on Kähler manifolds with two Kähler metrics. It is the gradient flow of the J -functional that appears in Chen's formula for the Mabuchi energy. We find a positivity condition in terms of the two metrics that is both necessary and sufficient for the convergence of the J-flow to a critical metric. We use this result to show that on manifolds with ample canonical bundle, the Mabuchi energy is proper on all Kähler classes in an open neighborhood of the canonical class defined by a positivity condition. This improves previous results of Chen and of the second author. We discuss the implications of this for the problem of the existence of constant-scalar-curvature Kähler metrics. We also study the singularities of the J-flow and, under certain conditions (which always hold for dimension 2) derive some estimates away from a subvariety. We discuss the conjectural remark of Donaldson that if the J-flow does not converge on a Kähler surface, then it should blow up over some curves of negative self-intersection.

Original languageEnglish (US)
Pages (from-to)210-229
Number of pages20
JournalCommunications on Pure and Applied Mathematics
Volume61
Issue number2
DOIs
StatePublished - Feb 1 2008
Externally publishedYes

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Singularity
Metric
Energy
Positivity
Constant Scalar Curvature
Self-intersection
Gradient Flow
Blow-up
Bundle
Sufficient
Converge
Curve
Necessary
Estimate
Class

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Mathematics(all)

Cite this

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abstract = "The J-flow of S. K. Donaldson and X. X. Chen is a parabolic flow on K{\"a}hler manifolds with two K{\"a}hler metrics. It is the gradient flow of the J -functional that appears in Chen's formula for the Mabuchi energy. We find a positivity condition in terms of the two metrics that is both necessary and sufficient for the convergence of the J-flow to a critical metric. We use this result to show that on manifolds with ample canonical bundle, the Mabuchi energy is proper on all K{\"a}hler classes in an open neighborhood of the canonical class defined by a positivity condition. This improves previous results of Chen and of the second author. We discuss the implications of this for the problem of the existence of constant-scalar-curvature K{\"a}hler metrics. We also study the singularities of the J-flow and, under certain conditions (which always hold for dimension 2) derive some estimates away from a subvariety. We discuss the conjectural remark of Donaldson that if the J-flow does not converge on a K{\"a}hler surface, then it should blow up over some curves of negative self-intersection.",
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On the convergence and singularities of the J-flow with applications to the Mabuchi energy. / Song, Jian; Weinkove, Ben.

In: Communications on Pure and Applied Mathematics, Vol. 61, No. 2, 01.02.2008, p. 210-229.

Research output: Contribution to journalArticle

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