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Positive mass theorem for the Yamabe problem on spin manifolds

Bernd Ammann·Emmanuel Humbert
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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. In the present article, we will give a proof for this fact which is considerably shorter than previous proofs. Our proof is a modification of Witten's argument, but no analysis on asymtotically flat spaces is needed.

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Bernd Ammann, Emmanuel Humbert. 2008-02-25. Positive mass theorem for the Yamabe problem on spin manifolds. https://arxiv.org/abs/math/0304043

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