Eguchi--Hanson metrics arising from Kahler--Einstein edge metrics

TL;DR

This paper constructs and analyzes a family of Kahler--Einstein edge metrics on Calabi--Hirzebruch manifolds, revealing that the Eguchi--Hanson metric naturally emerges as a Gromov--Hausdorff limit, and classifies all possible limits.

Contribution

It introduces a new family of Kahler--Einstein edge metrics on Calabi--Hirzebruch manifolds and characterizes their Gromov--Hausdorff limits, including the Eguchi--Hanson metric.

Findings

Eguchi--Hanson metric appears as a Gromov--Hausdorff limit.

Complete classification of Gromov--Hausdorff limits with diverse behaviors.

Resolution of a conjecture by Cheltsov--Rubinstein.

Abstract

Calabi--Hirzebruch manifolds are higher-dimensional generalizations of both the football and Hirzebruch surfaces. We construct a family of Kahler--Einstein edge metrics singular along two disjoint divisors on the Calabi--Hirzebruch manifolds and study their Gromov--Hausdorff limits when either cone angle tends to its extreme value. As a very special case, we show that the celebrated Eguchi--Hanson metric arises in this way naturally as a Gromov--Hausdorff limit. We also completely describe all other (possibly rescaled) Gromov--Hausdorff limits which exhibit a wide range of behaviors, resolving in this setting a conjecture of Cheltsov--Rubinstein. This gives a new interpretation of both the Eguchi--Hanson space and Calabi's Ricci flat spaces as limits of compact singular Einstein spaces.