The Kähler-Ricci flow on Hirzebruch surfaces

Details The Kähler-Ricci flow on Hirzebruch surfaces The Kähler-Ricci flow on Hirzebruch surfaces

2009-03-11

arxiv.org/abs/0903.1900v2

Mathematics

Differential Geometry

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

Scientific paper

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.

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