Geometric and Functional Analysis

, Volume 22, Issue 1, pp 240–265

Metric Flips with Calabi Ansatz


DOI: 10.1007/s00039-012-0151-1

Cite this article as:
Song, J. & Yuan, Y. Geom. Funct. Anal. (2012) 22: 240. doi:10.1007/s00039-012-0151-1


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 flowGromov-Hausdorff convergencesmall contractionflip

2010 Mathematics Subject Classification


Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA
  2. 2.Department of MathematicsJohns Hopkins UniversityBaltimoreUSA