The Sasaki Join and Admissible K\"ahler Constructions

TL;DR

This paper surveys a method combining Sasaki join and admissible K"ahler constructions to produce new extremal and constant scalar curvature Sasaki metrics, including Sasaki-Einstein metrics, with applications to the CR Yamabe problem.

Contribution

It introduces a novel method that merges Sasaki join and admissible K"ahler constructions to generate new geometric structures with specific curvature properties.

Findings

Constructed new extremal and constant scalar curvature Sasaki metrics.

Provided explicit solutions to the CR Yamabe problem.

Demonstrated non-uniqueness in solutions when the Yamabe invariant is positive.

Abstract

We give a survey of our recent work describing a method which combines the Sasaki join construction with the admissible K\"ahler construction of to obtain new extremal and new constant scalar curvature Sasaki metrics, including Sasaki-Einstein metrics. The constant scalar curvature Sasaki metrics also provide explicit solutions to the CR Yamabe problem. In this regard we give examples of the lack of uniqueness when the Yamabe invariant is positive.