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arXiv · 2201.10832

Moment polytopes on Sasaki manifolds and volume minimization

Akito Futaki
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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.

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Akito Futaki. 2022-01-26. Moment polytopes on Sasaki manifolds and volume minimization. https://arxiv.org/abs/2201.10832

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