Non-orderability and the contact Hofer norm

TL;DR

This paper explores the relationship between non-orderability in contact topology and the contact Hofer norm, providing new examples of non-orderable contact manifolds and establishing properties of the contact Hofer metric.

Contribution

It introduces a novel connection between non-orderability and the contact Hofer norm, expanding the class of known non-orderable contact manifolds and analyzing the contact Hofer metric's properties.

Findings

Many new examples of non-orderable contact manifolds including boundaries of Weinstein domains

Existence of contactomorphisms without translated points in certain manifolds

The contact Hofer metric is $ ext{C}^0$-continuous

Abstract

We relate non-orderability in contact topology to shortening in the contact Hofer norm. Combined with considerations of open books, this provides many new examples of non-orderable contact manifolds, including contact boundaries of subcritical Weinstein domains, and in particular the long-standing case of the standard $S^1 \times S^2.$ We also produce new examples of contact manifolds admitting contactomorphisms without translated points, provide obstructions to subcritical polarizations of symplectic manifolds, and establish a $\mathcal{C}^0$-continuity property of the contact Hofer metric.