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

Boyer, Charles¹ ; Huang, Hongnian¹ ; Legendre, Eveline²

¹ Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, United States
² Institut de Mathematiques de Toulouse, Université Paul Sabatier, Toulouse, France

Geometry & topology, Tome 22 (2018) no. 7, pp. 4205-4234.

Résumé

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.

Détail

DOI : 10.2140/gt.2018.22.4205

Classification : 53B99, 53CXX, 53DXX
Keywords: Futaki invariant, constant scalar curvature metrics, Duistermaat–Heckman theorem, equivariant localisation, Sasaki and Kahler geometry

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