Positive scalar curvature with skeleton singularities

Abstract

We study positive scalar curvature on the regular part of Riemannian manifolds with singular, uniformly Euclidean (\(L^\infty \)) metrics that consolidate Gromov’s scalar curvature polyhedral comparison theory and edge metrics that appear in the study of Einstein manifolds. We show that, in all dimensions, edge singularities with cone angles \(\le 2\pi \) along codimension-2 submanifolds do not affect the Yamabe type. In three dimensions, we prove the same for more general singular sets, which are allowed to stratify along 1-skeletons, exhibiting edge singularities (angles \(\le 2\pi \)) and arbitrary \(L^\infty \) isolated point singularities. We derive, as an application of our techniques, Positive Mass Theorems for asymptotically flat manifolds with analogous singularities.

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Notes

  1. 1.

    Which was recently confirmed by the first-named author, alongside a rigidity result conjectured by Gromov, using altogether different methods; see [13].

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Acknowledgements

The authors would like to thank Rick Schoen, Brian White, Rafe Mazzeo, Pengzi Miao, and Or Hershkovits for stimulating conversations on the subject of this paper, as well as Gerhard Huisken, Dan Lee, André Neves, Yuguang Shi, and Peter Topping for their interest in this work. The first author would like to thank ETH-FIM for their hospitality, during which part of this work was carried out. The second author would like to thank the Ric Weiland Graduate Fellowship at Stanford, which partially supported the early portion of this research.

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Correspondence to Christos Mantoulidis.

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Communicated by Thomas Schick.

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Li, C., Mantoulidis, C. Positive scalar curvature with skeleton singularities. Math. Ann. 374, 99–131 (2019). https://doi.org/10.1007/s00208-018-1753-1

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