The K\"ahler-Ricci flow on Hirzebruch surfaces

TL;DR

This paper studies the long-term geometric behavior of the Kahler-Ricci flow on Hirzebruch surfaces, showing it either collapses, shrinks, or contracts divisors, confirming a previous conjecture and extending results to higher dimensions.

Contribution

It proves the conjecture of Feldman-Ilmanen-Knopf regarding the flow's behavior on Hirzebruch surfaces and extends the analysis to higher-dimensional analogues.

Findings

Flow either shrinks to a point, collapses to , or contracts an exceptional divisor.

Behavior confirmed in the sense of Gromov-Hausdorff convergence.

Similar behavior observed on higher-dimensional analogues.

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.