Existence and Non-Existence of Constant Scalar Curvature and Extremal Sasaki Metrics

TL;DR

This paper investigates the existence and properties of constant scalar curvature and extremal Sasaki metrics, providing new examples and counterexamples that clarify the structure of the Sasaki-Reeb cone and its relation to these metrics.

Contribution

It proves the existence of CSC Sasaki metrics on certain manifolds, shows the Sasaki-Reeb cone can be disconnected or empty, and explores constant weighted scalar curvature Kähler metrics.

Findings

CSC Sasaki metrics exist on specific admissible manifolds.

The extremal Sasaki-Reeb cone can be disconnected or empty.

A non-empty extremal cone may lack CSC Sasaki metrics.

Abstract

We discuss the existence and non-existence of constant scalar curvature, as well as extremal, Sasaki metrics. We prove that the natural Sasaki-Boothby-Wang manifold over the admissible projective bundles over local products of non-negative CSC K\"ahler metrics, as described in https://link-springer-com.libproxy.unm.edu/article/10.1007/s00222-008-0126-x, always has a constant scalar curvature (CSC) Sasaki metric in its Sasaki-Reeb cone. Moreover, we give examples that show that the extremal Sasaki--Reeb cone, defined as the set of Sasaki--Reeb vector fields admitting a compatible extremal Sasaki metric, is not necessarily connected in the Sasaki--Reeb cone, and it can be empty even in the non-Gorenstein case. We also show by example that a non-empty extremal Sasaki--Reeb cone need not contain a (CSC) Sasaki metric which answers a question posed in https://mathscinet-ams-org.libproxy.unm.edu/mathscinet-getitem?mr=4420789. The paper also contains an appendix where we explore the existence of K\"ahler metrics of constant weighted scalar curvature, as defined in https://londmathsoc-onlinelibrary-wiley-com.libproxy.unm.edu/doi/full/10.1112/plms.12255, on admissible manifolds over local products of non-negative CSC K\"ahler metrics.