The Weinstein conjecture for iterated planar contact structures

TL;DR

This paper proves the Weinstein conjecture for a class of contact manifolds supported by iterated planar open book decompositions, expanding the understanding of Reeb dynamics in higher dimensions.

Contribution

It introduces the concepts of iterated planar Lefschetz fibrations and open books, and proves the Weinstein conjecture for contact manifolds with these structures.

Findings

Weinstein conjecture holds for contact manifolds with iterated planar open books.

Supports higher-dimensional cases ($2n+1$ dimensions).

Establishes new links between open book decompositions and Reeb dynamics.

Abstract

In this paper, we introduce the notions of an iterated planar Lefschetz fibration and an iterated planar open book decomposition and prove the Weinstein conjecture for contact manifolds supporting an open book that has iterated planar pages. For $n\geq 1$, we show that a $(2n+1)$-dimensional contact manifold $M$ supporting an iterated planar open book decomposition satisfies the Weinstein conjecture.