On the transverse scalar curvature of a compact Sasaki manifold

TL;DR

This paper extends the concept of stability and geometric analysis from Kähler to Sasaki geometry, demonstrating that the space of Sasaki metrics forms an infinite-dimensional symmetric space and that the transverse scalar curvature acts as a moment map.

Contribution

It adapts the stability framework of constant scalar curvature metrics from Kähler to Sasaki geometry with minimal modifications.

Findings

The space of Sasaki metrics is an infinite-dimensional symmetric space.

Transverse scalar curvature is a moment map of the strict contactomorphism group.

The stability notions from Kähler geometry apply to Sasaki geometry.

Abstract

We show that the standard picture regarding the notion of stability of constant scalar curvature metrics in K\"ahler geometry described by S.K. Donaldson, which involves the geometry of infinite-dimensional groups and spaces, can be applied to the constant scalar curvature metrics in Sasaki geometry with only few modification. We prove that the space of Sasaki metrics is an infinite dimensional symmetric space and that the transverse scalar curvature of a Sasaki metric is a moment map of the strict contactomophism group.