Quasiconformal contact foliations

TL;DR

This paper studies quasiconformal contact foliations, demonstrating they support invariant metrics, are characterized by $C^1$-equicontinuity, and satisfy a generalized Weinstein conjecture with bounds on closed leaves.

Contribution

It introduces the concept of quasiconformal contact foliations, characterizes them via $C^1$-equicontinuity, and proves a generalized Weinstein conjecture for these structures.

Findings

Existence of invariant metrics on quasiconformal contact foliations.

Characterization of such foliations by $C^1$-equicontinuity.

Validation of a generalized Weinstein conjecture for quasiconformal Reeb fields.

Abstract

We show that every quasiconformal contact foliation supports an invariant metric and characterise such foliations by the dynamical property of $C^1$-equicontinuity. We prove that a generalisation of the Weinstein conjecture holds for quasiconformal contact foliations, and provide a lower bound to the number of closed leaves. In particular, we show that the Weinstein conjecture holds for quasiconformal Reeb fields.