The discriminant and oscillation lengths for contact and Legendrian isotopies

TL;DR

This paper introduces new bi-invariant metrics on contactomorphism groups, explores their boundedness properties on various manifolds, and applies these metrics to Legendrian isotopies, revealing unbounded lengths in several cases.

Contribution

It defines the discriminant and oscillation metrics on contactomorphism groups, analyzes their boundedness, and applies these concepts to Legendrian isotopies, providing new tools for contact topology.

Findings

Discriminant metric is unbounded on R^{2n} x S^1 and RP^{2n+1}.

Discriminant metric is bounded on R^{2n+1} and S^{2n+1}.

Discriminant and oscillation lengths are unbounded for certain Legendrian isotopies.

Abstract

We define an integer-valued non-degenerate bi-invariant metric (the discriminant metric) on the universal cover of the identity component of the contactomorphism group of any contact manifold. This metric has a very simple geometric definition, based on the notion of discriminant points of contactomorphisms. Using generating functions we prove that the discriminant metric is unbounded for the standard contact structures on R^{2n} x S^1 and RP^{2n+1}. On the other hand we also show by elementary arguments that the discriminant metric is bounded for the standard contact structures on R^{2n+1} and S^{2n+1}. As an application of these results we get that the contact fragmentation norm is unbounded for R^{2n} x S^1 and RP^{2n+1}. By elaborating on the construction of the discriminant metric we then define a second integer-valued bi-invariant metric, that we call the discriminant oscillation metric. This second metric is non-degenerate if and only if the contact manifold is orderable in the sense of Eliashberg and Polterovich and, in this case, it is compatible with the partial order. Finally we define the discriminant and oscillation lengths of a Legendrian isotopy, and prove that they are unbounded for T^{\ast}B x S^1 for any closed manifold B, for RP^{2n+1} and for some 3-dimensional circle bundles.