Canonical Sasakian Metrics

Abstract

Let $M$ be a closed manifold of Sasaki type. A polarization of $M$ is defined by a Reeb vector field, and for one such, we consider the set of all Sasakian metrics compatible with it. On this space, we study the functional given by the squared $L^2$-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open.

36 pages, minor corrections made, example added

References in corpus (2)

• Sasakian Geometry, Hypersurface Singularities, and Einstein Metrics
• Holomorphicity and Walczak formula on Sasakian manifolds

Cited by in corpus (3)

• Uniqueness and examples of compact toric Sasaki-Einstein metrics
• Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds
• Einstein metrics and GIT stability