Sasakian geometry on sphere bundles II: Constant scalar curvature

In a previous paper [18] the authors employed the fiber join construction of Yamazaki [38] together with the admissible construction of Apostolov, Calderbank, Gauduchon, and Tønnesen-Friedman [2] to construct new extremal Sasaki metrics on odd dimensional sphere bundles over smooth projective algebraic varieties. In the present paper we continue this study by applying a recent existence theorem [14] that shows that under certain conditions one can always obtain a constant scalar curvature Sasaki metric in the Sasaki cone. Moreover, we explicitly describe this construction for certain sphere bundles of dimension 5 and 7.