Moment polytopes on Sasaki manifolds and volume minimization

TL;DR

This paper demonstrates that transverse coupled Kähler-Einstein metrics on toric Sasaki manifolds can be characterized as critical points of a volume functional, extending volume minimization techniques to the coupled setting under specific geometric conditions.

Contribution

It introduces a new approach to analyze transverse coupled Kähler-Einstein metrics via moment polytopes and extends volume minimization methods to the coupled case with Minkowski sum assumptions.

Findings

Transverse coupled Kähler-Einstein metrics are critical points of a volume functional.

Revisits moment polytopes under Reeb vector field changes.

Extends volume minimization to coupled metrics with Minkowski sum assumption.

Abstract

We show that transverse coupled K\"ahler-Einstein metrics on toric Sasaki manifolds arise as a critical point of a volume functional. As a preparation for the proof, we re-visit the transverse moment polytopes and contact moment polytopes under the change of Reeb vector fields. Then we apply it to a coupled version of the volume minimization by Martelli-Sparks-Yau. This is done assuming the Calabi-Yau condition of the K\"ahler cone, and the non-coupled case leads to a known existence result of a transverse K\"ahler-Einstein metric and a Sasaki-Einstein metric, but the coupled case requires an assumption related to Minkowski sum to obtain transverse coupled K\"ahler-Einstein metrics.