On effective uniformization of 2-sphere and the stability

Abstract

We provide a proof of effective uniformization for nearly round 2-spheres, utilizing an identity related to the third-order differential of the conformal factor. This identity is connected to the geometry of the embedded spacelike surface within the Minkowski lightcone. Additionally, we investigate the stability of the effective uniformization introduced by Klainerman and Szeftel. Our proof is based on a geometric insight: an isometric embedding of a round sphere into Euclidean space can be constructed using an orthogonal basis of the first eigenspace of the Laplacian operator, with the rectangular coordinates corresponding to the basis functions. By adopting these approaches, we simplify both the proofs of effective uniformization and its stability, while also refining the assumptions underlying both results.