Geometric and Functional Analysis

, Volume 22, Issue 1, pp 240-265

Metric Flips with Calabi Ansatz

  • Jian SongAffiliated withDepartment of Mathematics, Rutgers University
  • , Yuan YuanAffiliated withDepartment of Mathematics, Johns Hopkins University Email author 

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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.

Keywords and phrases

Kähler-Ricci flow Gromov-Hausdorff convergence small contraction flip

2010 Mathematics Subject Classification

53c55 53c44