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

TL;DR

This paper introduces a method to generate new constant scalar curvature Sasaki metrics from existing ones, explores their non-uniqueness, and discusses special cases like Sasaki Ricci solitons and Einstein metrics.

Contribution

It provides a general construction procedure for CSC Sasaki metrics, demonstrates non-uniqueness of CSC rays, and characterizes Sasaki Ricci solitons and Einstein metrics in specific cases.

Findings

Constructed new CSC Sasaki metrics from existing ones.

Showed CSC rays are often not unique on fixed CR or contact manifolds.

Identified a 2D subcone of Sasaki Ricci solitons and unique Sasaki-Einstein metrics when the first Chern class vanishes.

Abstract

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.