Two closed orbits for non-degenerate Reeb flows

We prove that every non-degenerate Reeb flow on a closed contact manifold M admitting a strong symplectic filling W with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic homology of W satisfies a mild condition. Under further assumptions, we establish the existence of two geometrically distinct closed orbits on any contact finite quotient of M. Several examples of such contact manifolds are provided, like displaceable ones, unit cosphere bundles, prequantisation circle bundles, Brieskorn spheres and toric contact manifolds. We also show that this condition on the equivariant symplectic homology is preserved by boundary connected sums of Liouville domains. As a byproduct of one of our applications, we prove a sort of Lusternik–Fet theorem for Reeb flows on the unit cosphere bundle of not rationally aspherical manifolds satisfying suitable additional assumptions.

On the Mean Euler Characteristic of Gorenstein Toric Contact Manifolds

We prove that the mean Euler characteristic of a Gorenstein toric contact manifold, that is, a good toric contact manifold with zero 1st Chern class, is equal to half the normalized volume of the corresponding toric diagram and give some applications. A particularly interesting one, obtained using a result of Batyrev and Dais, is the following: twice the mean Euler characteristic of a Gorenstein toric contact manifold is equal to the Euler characteristic of any crepant toric symplectic filling, that is, any toric symplectic filling with zero 1st Chern class.

Toric Contact Geometry in Arbitrary Codimension

We define toric contact manifolds in arbitrary codimension and give a description of such manifolds in terms of a kind of labelled polytope embedded into a grassmannian, analogous to the Delzant polytope of a toric symplectic manifold.

Contact blow up and cylindrical contact homology of toric contact manifolds of Reeb type

Let (V,?) be a toric contact manifold of Reeb type that is a prequantization of a toric symplectic manifold (M,?). A contact blow up of (V,?) is the prequantization of a symplectic blow up of (M,?). Thus, a contact blow up of (V,?) is a new toric contact manifold of Reeb type. In some special cases we are able to compute the cylindrical contact homology for the contact blowup using only the cylindrical contact homology of the contact manifold we started with.

Contact homology of good toric contact manifolds

In this paper we show that any good toric contact manifold has a well-defined cylindrical contact homology, and describe how it can be combinatorially computed from the associated moment cone. As an application, we compute the cylindrical contact homology of a particularly nice family of examples that appear in the work of Gauntlett et al. on Sasaki–Einstein metrics. We show in particular that these give rise to a new infinite family of non-equivalent contact structures on S2×S3 in the unique homotopy class of almost contact structures with vanishing first Chern class.

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

We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least five. 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 an existence result of E. Legendre [Toric geometry of convex quadrilaterals, J. Symplectic Geom. 9 (2011), 343–385], we show that a co-oriented compact toric contact 5-manifold whose moment cone has four 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 S2×S3 admitting two non-isometric and non-transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.