Twins in K{\"a}hler and Sasaki geometry

TL;DR

This paper introduces the concepts of weighted extremal Kähler and Sasaki twins, generalizing twinning phenomena in geometric structures and revealing multiple extremal rays in the Sasaki cone, especially in toric cases.

Contribution

It defines weighted extremal twins in Kähler and Sasaki geometry, extending previous twinning phenomena to broader classes and analyzing their implications in toric Sasaki structures.

Findings

Many twins appear in the weighted extremal setting.

Multiple extremal rays can exist in the Sasaki cone without isotopy changes.

The study focuses on the toric Sasaki case.

Abstract

We introduce the notions of weighted extremal K{\"a}hler twins together with the related notion of extremal Sasaki twins. In the K\"ahler setting this leads to a generalization of the twinning phenomenon appearing among LeBrun's strongly Hermitian solutions to the Einstein-Maxwell equations on the first Hirzebruch surface \cite{Leb16} to weighted extremal metrics on Hirzebruch surfaces in general. We discover that many twins appear and that this can be viewed in the Sasaki setting as a case where we have more than one extremal ray in the Sasaki cone even when we do not allow changes within the isotopy class. We also study extremal Sasaki twins directly in the Sasaki setting with a main focus on the toric Sasaki case.