# Positive mass theorem for the Yamabe problem on spin manifolds by Bernd Ammann, Emmanuel Humbert

Positive mass theorem for the Yamabe problem on spin manifolds (.dvi, .ps,.ps.gz oder .pdf)
GAFA 15 (2005), 567-576.

### Abstract

Let $(M,g)$ be a compact connected spin manifold of dimension $n\geq 3$ whose Yamabe invariant is positive. We assume that $(M,g)$ is locally conformally flat or that $n \in \{3,4,5\}$. According to a positive mass theorem of Witten, the constant term in the asymptotic development of the Green's function of the conformal Laplacian is positive if $(M,g)$ is not conformally equivalent to the sphere. Using Witten's argument, we give a very short proof of this fact. This simplifies considerably the proof of the Yamabe problem for spin manifolds.

### Mathematics Subject Classification

53C21 (Primary), 58E11, 53C27 (Secondary)

### Erratum

In the printed version a term in a local development of the Dirac operator is not complete, a term is missing.
Here is the corrected version: .dvi, .ps,.ps.gz oder .pdf
Here is the preprint version close to the printed version: .dvi, .ps,.ps.gz oder .pdf

### Keywords

Positive mass theorem, Yamabe problem, spin manifolds
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Bernd Ammann, Emmanuel Humbert,
The Paper was written on 01.04.2003
Last update of the www-page 21.12.2007