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 Kähler 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.