8 research outputs found
The persistence of the Chekanov-Eliashberg algebra
We apply the barcodes of persistent homology theory to the
Chekanov-Eliashberg algebra of a Legendrian submanifold to deduce displacement
energy bounds for arbitrary Legendrians. We do not require the full
Chekanov-Eliashberg algebra to admit an augmentation as we linearize the
algebra only below a certain action level. As an application we show that it is
not possible to -approximate a stabilized Legendrian by a Legendrian that
admits an augmentation.Comment: 29 pages, 4 figures; version accepted for publication in Selecta
Mathematica. This is a major revision with many fixes and improvements. The
constant in Theorem 1.1 has been improved. The theory of barcodes have been
properly introduced in the new Section 2 together with new related
terminology. The proof of Theorem 1.1 was rewritten in the new language and
given a greater level of detail
-limits of Legendrians and positive loops
We show that the image of a Legendrian submanifold under a homeomorphism that
is the -limit of a sequence of contactomorphisms is again Legendrian, if
the image of the submanifold is smooth. In proving this, we show that any
non-Legendrian submanifold of a contact manifold admits a positive loop and we
provide a parametric refinement of the Rosen--Zhang result on the degeneracy of
the Chekanov--Hofer--Shelukhin pseudo-norm for non-Legendrians.Comment: Added two references and Remark 2.
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