A hypergeometric proof that ${\sf Iso}$ is bijective

TL;DR

This paper presents an elementary proof demonstrating that the ${ m Iso}$ function is bijective, using an explicit hypergeometric function quotient, simplifying the understanding of its properties.

Contribution

It introduces a new proof of the bijectivity of ${ m Iso}$ using hypergeometric functions, providing a more elementary approach than previous methods.

Findings

${ m Iso}$ is bijective, as shown by the hypergeometric proof.

The proof simplifies understanding of the ${ m Iso}$ function's properties.

Explicit hypergeometric expression facilitates further analysis.

Abstract

We provide a short and elementary proof of the main technical result of the recent article "On the uniqueness of Clifford torus with prescribed isoperimetric ratio" by Thomas Yu and Jingmin Chen. The key of the new proof is an explicit expression of the central function (${\sf Iso}$, to be proved bijective) as a quotient of Gaussian hypergeometric functions.