An application of the Duistertmaat--Heckman Theorem and its extensions in Sasaki Geometry

TL;DR

This paper applies the Duistermaat-Heckman localization formula and its extensions to Sasaki geometry, providing explicit formulas for key functionals and analyzing their properties, including minimization and invariants.

Contribution

It introduces rational, explicit expressions for volume and curvature functionals in Sasaki geometry using advanced localization techniques, extending previous ideas.

Findings

The volume, scalar curvature, and Einstein-Hilbert functional are rational and explicit on the Sasaki cone.

The functionals are proper and attain their minima within the cone.

Existence of Reeb vector fields with vanishing transverse Futaki invariant in each Sasaki cone.

Abstract

Building on an idea laid out by Martelli--Sparks--Yau, we use the Duistermaat-Heckman localization formula and an extension of it to give rational and explicit expressions of the volume, the total transversal scalar curvature and the Einstein--Hilbert functional, seen as functionals on the Sasaki cone (Reeb cone). Studying the leading terms we prove they are all proper. Among consequences we get that the Einstein-Hilbert functional attains its minimal value and each Sasaki cone possess at least one Reeb vector field with vanishing transverse Futaki invariant.