G-Sasaki Manifolds and K-Energy

In this paper, we introduce a class of Sasaki manifolds, called G-Sasaki manifolds with a reductive G-group action on their Kahler cones. By proving the properness of K-energy on such manifolds, we obtain a sufficient and necessary condition for the existence of G-Sasaki-Einstein metrics. A similar result is also obtained for G-Sasaki-Ricci solitons. As an application, we construct many new examples of G-Sasaki-Ricci solitons by an established openness theorem.

Automated summary

This paper introduces a class of G-Sasaki manifolds with a reductive G-group action on their Kahler cones, proving the properness of K-energy on such manifolds and obtaining a sufficient and necessary condition for the existence of G-Sasaki-Einstein metrics. The results also apply to G-Sasaki-Ricci solitons, leading to the construction of many new examples through an established openness theorem.