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arXiv · 1004.2018

On the Kähler-Ricci flow near a Kähler-Einstein metric

Song Sun·Yuanqi Wang
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Abstract

On a Fano manifold, we prove that the Kahler-Ricci flow starting from a Kahler metric in the anti-canonical class which is sufficiently close to a Kahler-Einstein metric must converge in a polynomial rate to a Kahler-Einstein metric. The convergence can not happen in general if we study the flow on the level of Kahler potentials. Instead we exploit the interpretation of the Ricci flow as the gradient flow of Perelman's μfunctional. This involves modifying the Ricci flow by a canonical family of gauges. In particular, the complex structure of the limit could be different in general. The main technical ingredient is a Lojasiewicz type inequality for Perelman's μfunctional near a critical point.

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Song Sun, Yuanqi Wang. 2013-01-15. On the Kähler-Ricci flow near a Kähler-Einstein metric. https://arxiv.org/abs/1004.2018

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