Contact spheres and hyperk\"ahler geometry

A taut contact sphere on a 3-manifold is a linear 2-sphere of contact forms, all defining the same volume form. In the present paper we completely determine the moduli of taut contact spheres on compact left-quotients of SU(2) (the only closed manifolds admitting such structures). We also show that the moduli space of taut contact spheres embeds into the moduli space of taut contact circles. This moduli problem leads to a new viewpoint on the Gibbons-Hawking ansatz in hyperkahler geometry. The classification of taut contact spheres on closed 3-manifolds includes the known classification of 3-Sasakian 3-manifolds, but the local Riemannian geometry of contact spheres is much richer. We construct two examples of taut contact spheres on open subsets of 3-space with nontrivial local geometry; one from the Helmholtz equation on the 2-sphere, and one from the Gibbons-Hawking ansatz. We address the Bernstein problem whether such examples can give rise to complete metrics.Comment: 29 pages, v2: Large parts have been rewritten; previous Section 6 has been removed; new Section 5.2 on the Gibbons-Hawking ansatz; new Sections 6 and

Discontinuous symplectic capacities

We show that the spherical capacity is discontinuous on a smooth family of ellipsoidal shells. Moreover, we prove that the shell capacity is discontinuous on a family of open sets with smooth connected boundaries.Comment: We include generalizations to higher dimensions due to the unknown referee and Janko Latschev. We add examples of open sets with connected boundary on which the shell capacity is not continuous. 3rd and 4th version: minor changes, to appear in J. Fixed Point Theory App

Remarks on Legendrian Self-Linking

The Thurston-Bennequin invariant provides one notion of self-linking for any homologically-trivial Legendrian curve in a contact three-manifold. Here we discuss related analytic notions of self-linking for Legendrian knots in Euclidean space. Our definition is based upon a reformulation of the elementary Gauss linking integral and is motivated by ideas from supersymmetric gauge theory. We recover the Thurston-Bennequin invariant as a special case.Comment: 42 pages, many figures; v2: minor revisions, published versio

How to recognize a 4-ball when you see one

We apply the method of filling with holomorphic discs to a 4-dimensional symplectic cobordism with the standard contact 3-sphere as one convex boundary component. We establish the following dichotomy: either the cobordism is diffeomorphic to a ball, or there is a periodic Reeb orbit of quantifiably short period in the concave boundary of the cobordism. This allows us to give a unified treatment of various results concerning Reeb dynamics on contact 3-manifolds, symplectic fillability, the topology of symplectic cobordisms, symplectic nonsqueezing, and the nonexistence of exact Lagrangian surfaces in standard symplectic 4-space

Eliashberg's proof of Cerf's theorem

Following a line of reasoning suggested by Eliashberg, we prove Cerf's theorem that any diffeomorphism of the 3-sphere extends over the 4-ball. To this end we develop a moduli-theoretic version of Eliashberg's filling-with-holomorphic-discs method.Comment: 32 page

Brieskorn manifolds as contact branched covers of spheres

We show that Brieskorn manifolds with their standard contact structures are contact branched coverings of spheres. This covering maps a contact open book decomposition of the Brieskorn manifold onto a Milnor open book of the sphere.Comment: 8 pages, 1 figur

Algebraic Torsion in Contact Manifolds

We extract a nonnegative integer-valued invariant, which we call the "order of algebraic torsion", from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic torsion of order zero if and only if it is algebraically overtwisted (i.e. has trivial contact homology), and any contact 3-manifold with positive Giroux torsion has algebraic torsion of order one (though the converse is not true). We also construct examples for each nonnegative k of contact 3-manifolds that have algebraic torsion of order k but not k - 1, and derive consequences for contact surgeries on such manifolds. The appendix by Michael Hutchings gives an alternative proof of our cobordism obstructions in dimension three using a refinement of the contact invariant in Embedded Contact Homology.Comment: 53 pages, 4 figures, with an appendix by Michael Hutchings; v.3 is a final update to agree with the published paper, and also corrects a minor error that appeared in the published version of the appendi

Calabi-Yau cones from contact reduction

We consider a generalization of Einstein-Sasaki manifolds, which we characterize in terms both of spinors and differential forms, that in the real analytic case corresponds to contact manifolds whose symplectic cone is Calabi-Yau. We construct solvable examples in seven dimensions. Then, we consider circle actions that preserve the structure, and determine conditions for the contact reduction to carry an induced structure of the same type. We apply this construction to obtain a new hypo-contact structure on S^2\times T^3.Comment: 30 pages; v2: typos corrected, presentation improved, one reference added. To appear in Ann. Glob. Analysis and Geometr

Inequivalent contact structures on Boothby-Wang 5-manifolds

We consider contact structures on simply-connected 5-manifolds which arise as circle bundles over simply-connected symplectic 4-manifolds and show that invariants from contact homology are related to the divisibility of the canonical class of the symplectic structure. As an application we find new examples of inequivalent contact structures in the same equivalence class of almost contact structures with non-zero first Chern class.Comment: 27 pages; to appear in Math. Zeitschrif