Tingley's problem on uniform algebras

TL;DR

This paper solves Tingley's problem for uniform algebras by proving that a surjective isometry on their unit spheres extends to a real-linear isometry, marking a significant advancement in the geometry of Banach spaces of analytic functions.

Contribution

It provides the first positive solution to Tingley's problem for a Banach space of analytic functions, specifically uniform algebras.

Findings

Surjective isometry on unit spheres extends to real-linear isometry.

First positive solution for Tingley's problem in this context.

Advances understanding of geometric structure of uniform algebras.

Abstract

We prove that a surjective isometry between the unit spheres of two uniform algebras is extended to a surjective real-linear isometry between the uniform algebras. It provides the first positive solution for Tingley's problem on a Banach space of analytic functions.