On finite time Type I singularities of the Kähler–Ricci flow on compact Kähler surfaces

  • Charles Cifarelli

    Stony Brook University, Stony Brook, USA
  • Ronan J. Conlon

    The University of Texas at Dallas, Richardson, USA
  • Alix Deruelle

    Université Paris-Saclay, Orsay, France
On finite time Type I singularities of the Kähler–Ricci flow on compact Kähler surfaces cover
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Abstract

We show that the underlying complex manifold of a complete non-compact two-dimen­sional shrinking gradient Kähler–Ricci soliton with soliton metric with bounded scalar curvature whose soliton vector field has an integral curve along which is biholomorphic to either or to the blowup of this manifold at one point, and that the soliton metric  is toric. We also identify the corresponding soliton vector field in each case. Given these possibilities, we then prove a strong form of the Feldman–Ilmanen–Knopf conjecture for finite time Type I singularities of the Kähler–Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension.

Cite this article

Charles Cifarelli, Ronan J. Conlon, Alix Deruelle, On finite time Type I singularities of the Kähler–Ricci flow on compact Kähler surfaces. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1485