Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 1

Special Issue in Honor of Bernie Shiffman

Guest Editors: Yuan Yuan, Christopher Sogge, and Steven Morris Zelditch

Angle deformation of Kähler–Einstein edge metrics on Hirzebruch surfaces

Pages: 343 – 366

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n1.a11


Yanir A. Rubinstein (Department of Mathematics, University of Maryland, College Park, Md., U.S.A.)

Kewei Zhang (Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing, China)


We construct a family of Kähler–Einstein edge metrics on all Hirzebruch surfaces using the Calabi ansatz and study their angle deformation. This allows us to verify in some special cases a conjecture of Cheltsov–Rubinstein that predicts convergence towards a non-compact Calabi–Yau fibration in the small angle limit. We also give an example of a Kähler–Einstein edge metric whose edge singularity is rigid, answering a question posed by Cheltsov.


Kähler–Einstein edge metric, Calabi–Yau fiberation

2010 Mathematics Subject Classification

Primary 32Q20, 53C25. Secondary 32Q26.

Received 5 July 2020

Accepted 24 November 2020

Published 10 February 2022