2006/04/30 by Charles P. Boyer, Krzysztof Galicki, Santiago R. Simanca
Mathematics · #Curvature #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Holomorphic and Operator Theory #Invariant (physics) #Mathematical analysis #Mathematical physics #Mathematics #Polarization (electrochemistry) #Pure mathematics #Scalar curvature #Scalar field #Vector field #math.DG #msc:53C25
paper · pdf · doi:10.1007/s00220-008-0429-1
published as Commun.Math.Phys.279:705-733,2008 · 36 pages, minor corrections made, example added
arxiv created 2007/03/13 · openalex publication_date 2008/02/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for one such, we consider the set of all Sasakian metrics compatible with it. On this space, we study the functional given by the squared L2-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open.