Translated points on hypertight contact manifolds

TL;DR

This paper develops Rabinowitz Floer homology for hypertight contact manifolds and uses it to prove conjectures about translated points, invariant Reeb orbits, and the topology of contactomorphism loops.

Contribution

It introduces a new Floer homology framework for hypertight contact manifolds and applies it to resolve longstanding conjectures in contact topology.

Findings

Existence of translated points on hypertight contact manifolds

Presence of non-contractible Reeb orbits from positive loops

Validation of conjectures by Sandon and Mazzucchelli

Abstract

A contact manifold admittting a supporting contact form without contractible Reeb orbits is called hypertight. In this paper we construct a Rabinowitz Floer homology associated to an arbitrary supporting contact form for a hypertight contact manifold, and use this to prove versions of conjectures of Sandon and Mazzucchelli on the existence of translated points and invariant Reeb orbits, and to show that positive loops of contactomorphisms give rise to non-contractible Reeb orbits.