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arXiv · 0705.4048

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

D. H. Phong·Jian Song·Jacob Sturm·Ben Weinkove
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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 $\bar\partial^\dagger \bar\partial$ operator on smooth vector fields is bounded away from 0 along the flow, then the metrics converge exponentially fast in $C^\infty$ to a Kähler-Einstein metric.

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D. H. Phong, Jian Song, Jacob Sturm, Ben Weinkove. 2008-02-26. The Kähler-Ricci flow and the $\bar\partial$ operator on vector fields. https://arxiv.org/abs/0705.4048

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