Existence and non uniqueness of constant scalar curvature toric Sasaki metrics

TL;DR

This paper investigates the existence and uniqueness of constant scalar curvature toric Sasaki metrics on compact toric contact manifolds, revealing conditions for their existence and providing examples of multiple non-isometric metrics.

Contribution

It proves the existence of Reeb vector fields with vanishing transversal Futaki invariant and demonstrates the presence of multiple non-isometric constant scalar curvature metrics on certain manifolds.

Findings

Existence of Reeb vector fields with zero transversal Futaki invariant.

Finite number of constant scalar curvature metrics on specific 5-manifolds.

Examples of multiple non-isometric metrics on $S^2\times S^3$.

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