Kahler-Ricci flow with unbounded curvature

Huang, Shaochuang Yau Mathematical Sciences Center Tam, Luen-Fai The Chinese University of Hong Kong

Differential Geometry mathscidoc:1804.10002

Distinguished Paper Award in 2018

American Journal of Mathematics, 140, (1), 189-220, 2018.2
Let g(t) be a smooth complete solution to the Ricci flow on a noncompact manifold such that g(0) is Kahler. We prove that if |Rm(g(t))| is bounded by a/t for some a > 0, then g(t) is Kahler for t > 0. We prove that there is a constant a(n) > 0 depending only on n such that the following is true: Suppose g(t) is a smooth complete solution to the Kahler-Ricci flow on a non-compact n-dimensional complex manifold such that g(0) has nonnegative holomorphic bisectional curvature and |Rm(g(t))| ≤ a(n)/t, then g(t) has nonnegative holomorphic bisectional curvature for t > 0. These generalize the results by Wan-Xiong Shi. As applications, we prove that (i) any complete noncompact Kahler manifold with nonnegative complex sectional curvature and maximum volume growth is biholomorphic to C^n; and (ii) there is ε(n) > 0 depending only on n such that if (M^n,g_0) is a complete noncompact Kahler manifold of complex dimension n with nonnegative holomorphic bisectional curvature and maximum volume growth and if (1+ε(n))^{−1}h ≤ g_0 ≤ (1+ε(n))h for some Riemannian metric h with bounded curvature, then M is biholomorphic to C^n.
Kahler-Ricci flow, kahler condition, nonnegative bisectional curvature, uniformization
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@inproceedings{huang,2018kahler-ricci,
  title={Kahler-Ricci flow with unbounded curvature},
  author={Huang, Shaochuang, and Tam, Luen-Fai},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180429165053432699077},
  booktitle={American Journal of Mathematics},
  volume={140},
  number={1},
  pages={189-220},
  year={2018},
}
Huang, Shaochuang, and Tam, Luen-Fai. Kahler-Ricci flow with unbounded curvature. 2018. Vol. 140. In American Journal of Mathematics. pp.189-220. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180429165053432699077.
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