# Coupled Sasaki-Ricci solitons

## TL;DR

This paper extends the theory of coupled Kähler-Einstein metrics to the Sasaki setting, establishing isomorphisms, obstructions, and existence results for coupled Sasaki-Einstein metrics and solitons, especially in the toric case.

## Contribution

It introduces coupled Sasaki-Einstein metrics and solitons, proves an isomorphism of Lie algebras, extends obstructions, and demonstrates existence in the toric case.

## Key findings

- Isomorphism between transverse holomorphic vector fields and coupled basic functions.
- Extension of obstructions to existence of coupled Sasaki-Einstein metrics.
- Existence of toric coupled Sasaki-Einstein metrics when the first Chern class is positive.

## Abstract

Motivated by the study of coupled K\"ahler-Einstein metrics by Hultgren and Witt Nystr\"om and coupled K\"ahler-Ricci solitons by Hultgren, we study in this paper coupled Sasaki-Einstein metrics and coupled Sasaki-Ricci solitons. We first show an isomorphism between the Lie algebra of all transverse holomorphic vector fields and certain space of coupled basic functions related to coupled twisted Laplacians for basic functions, and obtain extensions of the well-known obstructions to the existence of K\"ahler-Einstein metrics to this coupled case. These results are reduced to coupled K\"ahler-Einstein metrics when the Sasaki structure is regular. Secondly we show the existence of toric coupled Sasaki-Einstein metrics when the basic first Chern class is positive extending the work of Hultgren.