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Metric Flips with Calabi Ansatz

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Abstract

We study the limiting behavior of the Kähler–Ricci flow on \({{\mathbb{P}(\mathcal{O}_{\mathbb{P}^n} \oplus \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus(m+1)})}}\) for m, n ≥ 1, assuming the initial metric satisfies the Calabi symmetry. We show that the flow either shrinks to a point, collapses to \({{\mathbb{P}^n}}\) or contracts a subvariety of codimension m + 1 in the Gromov–Hausdorff sense. We also show that the Kähler–Ricci flow resolves a certain type of cone singularities in the Gromov–Hausdorff sense.

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, New Brunswick, NJ, 08902, USA

    Jian Song

  2. Department of Mathematics, Johns Hopkins University, Baltimore, MD, 21218, USA

    Yuan Yuan

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  1. Jian Song
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  2. Yuan Yuan
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Correspondence to Yuan Yuan.

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The first named author is supported in part by National Science Foundation grant DMS-0847524 and a Sloan Foundation Fellowship.

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Song, J., Yuan, Y. Metric Flips with Calabi Ansatz. Geom. Funct. Anal. 22, 240–265 (2012). https://doi.org/10.1007/s00039-012-0151-1

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