The Kähler-Ricci flow on Hirzebruch surfaces

Abstract

We investigate the metric behavior of the Kahler-Ricci flow on the Hirzebruch surfaces, assuming the initial metric is invariant under a maximal compact subgroup of the automorphism group. We show that, in the sense of Gromov-Hausdorff, the flow either shrinks to a point, collapses to ${\mathbb{P}}^1$ or contracts an exceptional divisor, confirming a conjecture of Feldman-Ilmanen-Knopf. We also show that similar behavior holds on higher-dimensional analogues of the Hirzebruch surfaces.

27 pages, v2 published version to appear in J. Reine Angew. Math

References in corpus (1)

• On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow

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• On the collapsing rate of the Kähler-Ricci flow with finite-time singularity
• Convergence of general inverse ${\sigma }_k$-flow on Kähler manifolds with Calabi Ansatz
• Finite time extinction of the Kahler-Ricci flow