A note on a rigidity estimate for degree $\pm 1$ conformal maps on $\mathbb{S}^2$

TL;DR

This paper offers a concise alternative proof for a rigidity estimate concerning degree ±1 conformal maps on the 2-sphere, specifically focusing on Möbius transformations, building on prior work by Mantel, Muratov, and Simon.

Contribution

It provides a simplified proof of an existing rigidity estimate for degree ±1 conformal maps on the sphere, enhancing understanding of their geometric properties.

Findings

Alternative proof of the rigidity estimate

Clarification of the geometric structure of conformal maps

Potential implications for conformal geometry studies

Abstract

In this note we present a short alternative proof of an estimate obtained by A.B.-Mantel, C.B. Muratov and T.M. Simon in [3] regarding the rigidity of degree {$\pm 1$} conformal maps of $\mathbb{S}^2$, i.e. its M\"obius transformations.