Un th\'eor\`eme de la masse positive pour le probl\`eme de Yamabe en dimension paire

TL;DR

This paper provides an elementary proof of a positive mass theorem for the Yamabe problem on even-dimensional, conformally flat manifolds with positive scalar curvature, extending previous results without topological restrictions.

Contribution

It introduces a new elementary proof using differential forms for the positive mass theorem in even dimensions, avoiding spin or topological assumptions.

Findings

Positive mass theorem holds for even-dimensional conformally flat manifolds.

The proof does not require spin structures or topological conditions.

The constant term in the Green function's development is positive under given conditions.

Abstract

Let $(M,g)$ be a compact conformally flat manifold of dimension $n\geq4$ with positive scalar curvature. According to a positive mass theorem by Schoen and Yau, the constant term in the development of the Green function of the conformal Laplacian is positive if $(M,g)$ is not conformally equivalent to the sphere. On spin manifolds, there is an elementary proof of this fact by Ammann and Humbert, based on a proof of Witten. Using differential forms instead of spinors, we give an elementary proof on even dimensional manifolds, without any other topological assumption.