G-Sasaki manifolds and K-energy

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In this paper, we introduce a class of Sasaki manifolds with a reductive G-group action, called G-Sasaki manifolds. By reducing K-energy to a functional defined on a class of convex functions on a moment polytope, we give a criterion for the properness of K-energy. In particular, we deduce a sufficient and necessary condition related to the polytope for the existence of transverse G-Sasaki Einstein metrics. A similar result is also obtained for transverse G-Sasaki Ricci solitons. As an application, we construct several examples of G-Sasaki Ricci solitons by an established openness theorem for transverse G-Sasaki Ricci solitons.