# Ambient constructions for Sasakian $\eta$-Einstein manifolds

## TL;DR

This paper develops explicit formulas for ambient constructions of CR invariants on Sasakian $ ext{η}$-Einstein manifolds and studies the variations of total $Q$-prime curvature in this context.

## Contribution

It provides explicit ambient formulas for CR invariants on Sasakian $ ext{η}$-Einstein manifolds, facilitating their analysis and variation properties.

## Key findings

- Explicit formulas for CR invariant objects in Sasakian $ ext{η}$-Einstein manifolds.
- Analysis of the first and second variation of total $Q$-prime curvature.
- Enhanced understanding of CR invariants in special geometric settings.

## Abstract

The theory of ambient spaces is useful to define CR invariant objects, such as CR invariant powers of the sub-Laplacian, the $P$-prime operators, and $Q$-prime curvature. However in general, it is difficult to write down these objects in terms of the Tanaka-Webster connection. In this paper, we give those explicit formulas for CR manifolds satisfying an Einstein condition, called Sasakian $\eta$-Einstein manifolds. As an application, we study properties of the first and the second variation of the total $Q$-prime curvature at Sasakian $\eta$-Einstein manifolds.