Conformal Powers of the Laplacian via Stereographic Projection
C. Robin Graham
TL;DR
This paper provides a direct conformal-cut derivation of Branson's factorization for the conformally invariant operator on the sphere with principal part $\Delta_S^k$, by relating it to the Euclidean operator $\Delta^k$ through stereographic projection. The method introduces weight-shifting operators $M^w$ and $M_w$ to transported functions and reduces the problem to an explicit Euclidean operator identity that can be proven by induction using commutator relations. The key result is the operator identity on $\mathbb{R}^n$ that yields the sphere factors $\Delta_S - c_j$, with constants $c_j=(\tfrac{n}{2}+j-1)(\tfrac{n}{2}-j)$, thereby recovering Branson’s formula from the $k=1$ Yamabe case. This establishes a transparent, constructive bridge between flat and curved conformal geometries and clarifies the role of stereographic projection in generating Branson’s invariant operators for all $k$.
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.