Existence and non uniqueness of constant scalar curvature toric Sasaki metrics

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

2010-04-20

arxiv.org/abs/1004.3461v1

Compositio Mathematica 147 (2011), pp. 1613-1634

Mathematics

Differential Geometry

22 pages, no figure.

Scientific paper

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 $S^2\times S^3$ admitting two non isometric and non transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.

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