224 research outputs found
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
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
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
Weak and strong fillability of higher dimensional contact manifolds
For contact manifolds in dimension three, the notions of weak and strong
symplectic fillability and tightness are all known to be inequivalent. We
extend these facts to higher dimensions: in particular, we define a natural
generalization of weak fillings and prove that it is indeed weaker (at least in
dimension five),while also being obstructed by all known manifestations of
"overtwistedness". We also find the first examples of contact manifolds in all
dimensions that are not symplectically fillable but also cannot be called
overtwisted in any reasonable sense. These depend on a higher-dimensional
analogue of Giroux torsion, which we define via the existence in all dimensions
of exact symplectic manifolds with disconnected contact boundary.Comment: 68 pages, 5 figures. v2: Some attributions clarified, and other minor
edits. v3: exposition improved using referee's comments. Published by Invent.
Mat
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