Conformal Powers of the Laplacian via Stereographic Projection

TL;DR

This paper presents a novel derivation of Branson's factorization formula for conformally invariant powers of the Laplacian on the sphere, using stereographic projection to relate Euclidean and spherical operators.

Contribution

It introduces a new derivation method for Branson's formula by leveraging stereographic projection and known Euclidean conformally invariant operators.

Findings

Derivation of Branson's formula from Euclidean space operators

Connection between Euclidean and spherical conformally invariant operators

Simplified understanding of conformal powers of the Laplacian

Abstract

A new derivation is given of Branson's factorization formula for the conformally invariant operator on the sphere whose principal part is the k-th power of the scalar Laplacian. The derivation deduces Branson's formula from knowledge of the corresponding conformally invariant operator on Euclidean space (the k-th power of the Euclidean Laplacian) via conjugation by the stereographic projection mapping.