The Sasaki Join, Hamiltonian 2-forms, and Sasaki-Einstein Metrics

By combining the join construction from Sasakian geometry with the Hamiltonian 2-form construction from K\"ahler geometry, we recover Sasaki-Einstein metrics discovered by physicists. Our geometrical approach allows us to give an algorithm for computing the topology of these Sasaki-Einstein manifolds. In particular, we explicitly compute the cohomology rings for several cases of interest and give a formula for homotopy equivalence in one particular 7-dimensional case. We also show that our construction gives at least a two dimensional cone of both Sasaki-Ricci solitons and extremal Sasaki metrics.Comment: 38 pages, paragraph added to introduction and Proposition 4.1 added, Proposition 4.15 corrected, Remark 5.5 added, and explanation for irregular Sasaki-Einstein structures expanded. Reference adde

The Sasaki Join, Hamiltonian 2-forms, and Constant Scalar Curvature

We describe a general procedure for constructing new Sasaki metrics of constant scalar curvature from old ones. Explicitly, we begin with a regular Sasaki metric of constant scalar curvature on a 2n+1-dimensional compact manifold M and construct a sequence, depending on four integer parameters, of rays of constant scalar curvature (CSC) Sasaki metrics on a compact Sasaki manifold of dimension $2n+3$. We also give examples which show that the CSC rays are often not unique on a fixed strictly pseudoconvex CR manifold or a fixed contact manifold. Moreover, it is shown that when the first Chern class of the contact bundle vanishes, there is a two dimensional subcone of Sasaki Ricci solitons in the Sasaki cone, and a unique Sasaki-Einstein metric in each of the two dimensional sub cones.Comment: 32 pages. A gap in the argument of applying the admissibility conditions to irregular Sasakian structures is filled. Some minor corrections and additions are also made. This is the final version which will appear in the Journal of Geometric Analysis. It also encorporates much from our paper arXiv:1309.706

Iterated S3S^3 Sasaki Joins and Bott Orbifolds

We present a categorical relationship between iterated $S^3$ Sasaki-joins and Bott orbifolds. Then we show how to construct smooth Sasaki-Einstein (SE) structures on the iterated joins. These become increasingly complicated as dimension grows. We give an explicit construction of (infinitely many) smooth SE structures up through dimension eleven, and conjecture the existence of smooth SE structures in all odd dimensions.Comment: 19 pages, Paper submitted to the upcoming conference {\it AMAZER: Analysis of Monge-Amp\`ere, a tribute to Ahmed Zeriahi} at the Institute of Mathematics of Toulouse (June 202

The S^3_\bfw Sasaki Join Construction

The main purpose of this work is to generalize the S^3_\bfw Sasaki join construction M\star_\bfl S^3_\bfw described in \cite{BoTo14a} 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 (c_1(\cald)=0) we construct a polynomial whose coeffients are linear in the components of \bfw and whose unique root in the interval $(1,\infty)$ 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 \cite{ApCa18} of Apostolov and Calderbank as well as the relation with K-stability.Comment: 34 pages; An incorrect statement of Proposition 2.21 was corrected. This has no effect on the rest of the pape