Positive mass theorem for the Paneitz-Branson operator

TL;DR

This paper proves a positive mass theorem for the Paneitz-Branson operator on compact Riemannian manifolds, showing the constant term in its Green function is positive unless the manifold is conformally equivalent to the sphere.

Contribution

It establishes a positive mass theorem for the Paneitz-Branson operator, extending concepts from spin geometry to conformal geometry.

Findings

Constant term in Green function is positive unless manifold is conformally sphere.

The result generalizes positive mass theorems to fourth-order conformally invariant operators.

Proof is inspired by positive mass theorem techniques on spin manifolds.

Abstract

We prove that under suitable assumptions, the constant term in the Green function of the Paneitz-Branson operator on a compact Riemannian manifold $(M,g)$ is positive unless $(M,g)$ is conformally diffeomophic to the standard sphere. The proof is inspired by the positive mass theorem on spin manifolds by Ammann-Humbert.