Vol. 293, No. 2, 2018

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ISSN: 0030-8730
Scalar curvature and singular metrics

Yuguang Shi and Luen-Fai Tam

Vol. 293 (2018), No. 2, 427–470
DOI: 10.2140/pjm.2018.293.427
Abstract

Let Mn, n 3, be a compact differentiable manifold with nonpositive Yamabe invariant σ(M). Suppose g0 is a continuous metric with volume V (M,g0) = 1, smooth outside a compact set Σ, and is in Wloc1,p for some p > n. Suppose the scalar curvature of g0 is at least σ(M) outside Σ. We prove that g0 is Einstein outside Σ if the codimension of Σ is at least 2. If in addition, g0 is Lipschitz then g0 is smooth and Einstein after a change of the smooth structure. If Σ is a compact embedded hypersurface, g0 is smooth up to Σ from two sides of Σ, and if the difference of the mean curvatures along Σ at two sides of Σ has a fixed appropriate sign, then g0 is also Einstein outside  Σ. For manifolds with dimension between 3 and 7, without a spin assumption we obtain a positive mass theorem on an asymptotically flat manifold for metrics with a compact singular set of codimension at least 2.

Keywords
Yamabe invariants, positive mass theorems, singular metrics
Mathematical Subject Classification 2010
Primary: 53C20
Secondary: 83C99
Milestones
Received: 1 December 2016
Revised: 4 May 2017
Accepted: 11 September 2017
Published: 23 November 2017
Authors
Yuguang Shi
Key Laboratory of Pure and Applied Mathematics
School of Mathematical Sciences
Peking University
Beijing
China
Luen-Fai Tam
Institute of Mathematical Sciences and Department of Mathematics
Chinese University of Hong Kong
Shatin
China