$G$-Sasaki manifolds and K-energy

TL;DR

This paper introduces $G$-Sasaki manifolds with reductive group actions, providing criteria for the existence of special metrics via a reduction of K-energy, and constructs examples of Ricci solitons.

Contribution

It defines $G$-Sasaki manifolds, reduces K-energy to convex functions on polytopes, and establishes conditions for transverse $G$-Sasaki Einstein metrics and Ricci solitons.

Findings

Criterion for properness of K-energy in terms of convex functions.

Necessary and sufficient conditions for existence of transverse $G$-Sasaki Einstein metrics.

Construction of examples of $G$-Sasaki Ricci solitons.

Abstract

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.