Volume 22, issue 7 (2018)

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

Charles P Boyer, Hongnian Huang and Eveline Legendre

Geometry & Topology 22 (2018) 4205–4234

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.

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Futaki invariant, constant scalar curvature metrics, Duistermaat–Heckman theorem, equivariant localisation, Sasaki and Kahler geometry
Mathematical Subject Classification 2010
Primary: 53B99, 53CXX, 53DXX
Received: 28 August 2017
Accepted: 28 June 2018
Published: 6 December 2018
Proposed: Simon Donaldson
Seconded: Tobias H Colding, Jean-Pierre Otal
Charles P Boyer
Department of Mathematics and Statistics
University of New Mexico
Albuquerque, NM
United States
Hongnian Huang
Department of Mathematics and Statistics
University of New Mexico
Albuquerque, NM
United States
Eveline Legendre
Institut de Mathematiques de Toulouse
Université Paul Sabatier