47 research outputs found
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
Tightness in contact metric 3-manifolds
This paper begins the study of relations between Riemannian geometry and
global properties of contact structures on 3-manifolds. In particular we prove
an analog of the sphere theorem from Riemannian geometry in the setting of
contact geometry. Specifically, if a given three dimensional contact manifold
(M,\xi) admits a complete compatible Riemannian metric of positive 4/9-pinched
curvature then the underlying contact structure \xi is tight; in particular,
the contact structure pulled back to the universal cover is the standard
contact structure on S^3. We also describe geometric conditions in dimension
three for \xi to be universally tight in the nonpositive curvature setting.Comment: 29 pages. Added the sphere theorem, removed high dimensional material
and an alternate approach to the three dimensional tightness radius estimate
The Minimal Length of a Lagrangian Cobordism between Legendrians
To investigate the rigidity and flexibility of Lagrangian cobordisms between
Legendrian submanifolds, we investigate the minimal length of such a cobordism,
which is a -dimensional measurement of the non-cylindrical portion of the
cobordism. Our primary tool is a set of real-valued capacities for a Legendrian
submanifold, which are derived from a filtered version of Legendrian Contact
Homology. Relationships between capacities of Legendrians at the ends of a
Lagrangian cobordism yield lower bounds on the length of the cobordism. We
apply the capacities to Lagrangian cobordisms realizing vertical dilations
(which may be arbitrarily short) and contractions (whose lengths are bounded
below). We also study the interaction between length and the linking of
multiple cobordisms as well as the lengths of cobordisms derived from
non-trivial loops of Legendrian isotopies.Comment: 33 pages, 9 figures. v2: Minor corrections in response to referee
comments. More general statement in Proposition 3.3 and some reorganization
at the end of Section
Birational cobordism invariance of uniruled symplectic manifolds
A symplectic manifold is called {\em (symplectically) uniruled}
if there is a nonzero genus zero GW invariant involving a point constraint. We
prove that symplectic uniruledness is invariant under symplectic blow-up and
blow-down. This theorem follows from a general Relative/Absolute correspondence
for a symplectic manifold together with a symplectic submanifold. A direct
consequence is that symplectic uniruledness is a symplectic birational
invariant. Here we use Guillemin and Sternberg's notion of cobordism as the
symplectic analogue of the birational equivalence.Comment: To appear in Invent. Mat