An application of the Duistermaat–Heckman theorem and its extensions in Sasaki geometry

Abstract

Building on an idea laid out by Martelli, Sparks and Yau (2008), 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 thereof we get that the Einstein–Hilbert functional attains its minimal value and each Sasaki cone possesses at least one Reeb vector field with vanishing transverse Futaki invariant.