The Minimal Length of a Lagrangian Cobordism between Legendrians

TL;DR

This paper introduces capacities derived from Legendrian Contact Homology to measure the minimal length of Lagrangian cobordisms between Legendrians, revealing new bounds and interactions related to cobordism length and Legendrian linking.

Contribution

It develops a set of capacities from Legendrian Contact Homology to establish lower bounds on Lagrangian cobordism length, advancing understanding of Legendrian cobordism rigidity and flexibility.

Findings

Capacities provide lower bounds on cobordism length.

Vertical dilations can have arbitrarily short cobordisms.

Contractions have a bounded below length.

Abstract

To investigate the rigidity and flexibility of Lagrangian cobordisms between Legendrian submanifolds, we investigate the minimal length of such a cobordism, which is a $1$-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.