The Sasaki join and admissible Kähler constructions

We give a survey of our recent work (Boyer and Tønnesen-Friedman (2013) [50], [51] and [30], Boyer and Tønnesen-Friedman (2014) [33], [29] and [36]) describing a method which combines the Sasaki join construction of Boyer et al. (2007) [31] with the admissible Kähler construction of Apostolov et al. (2006, 2004, 2008) [26], [27], [14] and [25] 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 λ(M)λ(M) is positive.