# A Reeb flow on the three-sphere without a disk-like global surface of section

## TL;DR

This paper constructs specific Reeb flows on the standard tight three-sphere that lack disk-like global surfaces of section, using integrable systems and a connected sum approach, challenging previous assumptions about their existence.

## Contribution

It introduces a novel construction of Reeb flows without disk-like global surfaces of section on the three-sphere, expanding understanding of their dynamical properties.

## Key findings

- Reeb flows without disk-like global surfaces of section exist on the three-sphere
- Constructed using integrable systems and connected sum techniques
- Demonstrates limitations of existing global surface of section theory

## Abstract

We show that there are Reeb flows on the standard, tight three-sphere that do not admit global surfaces of section with one boundary component. In particular, the Reeb flows that we construct do not admit disk-like global surfaces of section. These Reeb flows are constructed using integrable systems, and a connected sum construction that extends the integrable system.