On the min-max width of unit volume three-spheres

TL;DR

This paper explores the maximum width of unit volume three-spheres within a conformal class, examining the properties of maximising metrics and their behavior under conformal changes using min-max theory.

Contribution

It provides new insights into the conformal geometry of three-spheres and characterizes the extremal metrics related to the width in the context of min-max theory.

Findings

Identifies conditions for maximum width in conformal classes

Describes the structure of extremal metrics

Analyzes the behavior of width under conformal deformations

Abstract

How large can be the width of Riemannian three-spheres of the same volume in the same conformal class? If a maximum value is attained, how does a maximising metric look like? What happens as the conformal class changes? In this paper, we investigate these and other related questions, focusing on the context of Simon-Smith min-max theory.