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On the KhlerRicci flow near a KhlerEinstein metric

SONG SUN Yuanqi Wang

Differential Geometry mathscidoc:1912.43455

Journal fr die reine und angewandte Mathematik (Crelles Journal), 2015, (699), 143-158
On a Fano manifold, we prove that the KhlerRicci flow starting from a Khler metric in the anti-canonical class which is sufficiently close to a KhlerEinstein metric must converge in a polynomial rate to a KhlerEinstein metric. The convergence cannot happen in general if we study the flow on the level of Khler potentials. Instead we exploit the interpretation of the Ricci flow as the gradient flow of Perelman's functional. 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 functional near a critical point.
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@inproceedings{songon,
  title={On the KhlerRicci flow near a KhlerEinstein metric},
  author={SONG SUN, and Yuanqi Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221114540168579015},
  booktitle={Journal fr die reine und angewandte Mathematik (Crelles Journal)},
  volume={2015},
  number={699},
  pages={143-158},
}
SONG SUN, and Yuanqi Wang. On the KhlerRicci flow near a KhlerEinstein metric. Vol. 2015. In Journal fr die reine und angewandte Mathematik (Crelles Journal). pp.143-158. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221114540168579015.
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