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Taming 3-manifolds using scalar curvature

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In this paper we address the issue of uniformly positive scalar curvature on noncompact 3-manifolds. In particular we show that the Whitehead manifold lacks such a metric, and in fact that \({\mathbb{R}^3}\) is the only contractible noncompact 3-manifold with a metric of uniformly positive scalar curvature. We also describe contractible noncompact manifolds of higher dimension exhibiting this curvature phenomenon. Lastly we characterize all connected oriented 3-manifolds with finitely generated fundamental group allowing such a metric.

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Correspondence to Shmuel Weinberger.

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Research partially supported by NSF Grants DMS-0405867, DMS-0805913 and DMS-0600216.

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Chang, S., Weinberger, S. & Yu, G. Taming 3-manifolds using scalar curvature. Geom Dedicata 148, 3–14 (2010).

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Mathematics Subject Classification (2000)