The 𝑆³_𝒘 Sasaki join construction

Abstract

The main purpose of this work is to generalize the 𝑆³_𝒘 Sasaki join construction 𝑀 ⋆_𝒍 𝑆³_𝒘 described in the authors' 2016 paper when the Sasakian structure on 𝑀 is regular, to the general case where the Sasakian structure is only quasi-regular. This gives one of the main results, Theorem 3.2, which describes an inductive procedure for constructing Sasakian metrics of constant scalar curvature. In the Gorenstein case (𝑐₁(𝒟) = 0) we construct a polynomial whose coeffients are linear in the components of 𝒘 and whose unique root in the interval (1, ∞) completely determines the Sasaki–Einstein metric. In the more general case we apply our results to prove that there exists infinitely many smooth 7-manifolds each of which admit infinitely many inequivalent contact structures of Sasaki type admitting constant scalar curvature Sasaki metrics (see Corollary 6.15). We also discuss the relationship with a recent paper of Apostolov and Calderbank as well as the relation with K-stability.