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

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

2007-05-28

arxiv.org/abs/0705.4048v2

Mathematics

Differential Geometry

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

Scientific paper

The limiting behavior of the normalized K\"ahler-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 $\bar\partial^\dagger \bar\partial$ operator on smooth vector fields is bounded away from 0 along the flow, then the metrics converge exponentially fast in $C^\infty$ to a K\"ahler-Einstein metric.

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