Perfect Reeb flows and action-index relations

TL;DR

This paper investigates perfect Reeb flows with vanishing contact homology differential, establishing bounds on closed orbits and action-index relations, and characterizes perfect flows on the standard contact three-sphere.

Contribution

It provides new bounds and resonance relations for perfect Reeb flows and characterizes such flows on the standard contact three-sphere.

Findings

Upper bounds on simple closed Reeb orbits on various contact manifolds

Action-index resonance relations for the standard contact sphere

Characterization of perfect Reeb flows on the standard contact three-sphere

Abstract

We study non-degenerate Reeb flows arising from perfect contact forms, i.e., the forms with vanishing contact homology differential. In particular, we obtain upper bounds on the number of simple closed Reeb orbits for such forms on a variety of contact manifolds and certain action-index resonance relations for the standard contact sphere. Using these results, we reprove a theorem due to Bourgeois, Cieliebak and Ekholm characterizing perfect Reeb flows on the standard contact three-sphere as non-degenerate Reeb flows with exactly two simple closed orbits.