Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 2

Moment polytopes on Sasaki manifolds and volume minimization

Pages: 487 – 513

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n2.a3


Akito Futaki (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)


We show that transverse coupled Kähler–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 Kähler cone, and the non-coupled case leads to a known existence result of a transverse Kähler–Einstein metric and a Sasaki–Einstein metric, but the coupled case requires an assumption related to Minkowski sum to obtain transverse coupled Kähler–Einstein metrics.

Received 3 February 2022

Accepted 18 July 2022

Published 7 April 2023