The Sasaki Cone and Extremal Sasakian Metrics

TL;DR

This paper explores the structure of the Sasaki cone on manifolds, introduces an energy functional to identify extremal Reeb vectors, and provides examples and results related to Sasaki extremal metrics, including a proof of a conjecture in dimension five.

Contribution

It introduces an energy functional over the Sasaki cone, characterizes strongly extremal Reeb vectors, and constructs examples of manifolds with specific properties of their Sasaki cones.

Findings

Sasaki cone can coincide with the extremal set in certain manifolds.

Examples of five-dimensional manifolds with extremal Sasaki metrics are constructed.

A conjecture of Orlik on homology torsion in links of singularities is proved in dimension five.

Abstract

We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold. We introduce an energy functional over the cone, and use its critical points to single out the strongly extremal Reeb vectors fields. Should one such vector field be a member of the extremal set, the scalar curvature of a Sasaki extremal metric representing it would have the smallest $L^2$-norm among all Sasakian metrics of fixed volume that can represent vector fields in the cone. We use links of isolated hypersurface singularities to produce examples of manifolds of Sasaki type, many of these in dimension five, whose Sasaki cone coincides with the extremal set, and examples where the extremal set is empty. We end up by proving that a conjecture of Orlik concerning the torsion of the homology groups of these links holds in the five dimensional case.