Existence and non-uniqueness of constant scalar curvature toric Sasaki metrics

DOI10.1112/S0010437X1100529XzbMATH Open1237.53039arXiv1004.3461MaRDI QIDQ3094105

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Publication date: 21 October 2011

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Abstract: We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using existence result of [25], we show that a co-oriented compact toric contact 5-manifold whose moment cone has 4 facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on $S^{2}imes{S}^3$ admitting two non isometric and non transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.

Full work available at URL: https://arxiv.org/abs/1004.3461

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On the transverse scalar curvature of a compact Sasaki manifold ⋮ Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds ⋮ The Sasaki join, Hamiltonian 2-forms, and constant scalar curvature ⋮ An application of the Duistermaat-Heckman theorem and its extensions in Sasaki geometry

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