SCALAR CURVATURE AND SINGULAR METRICS

Let M-n, n >= 3, be a compact differentiable manifold with nonpositive Yamabe invariant sigma(M). Suppose g(0) is a continuous metric with volume V (M, g(0)) = 1, smooth outside a compact set Sigma, and is in W-loc(1,p) for some p > n. Suppose the scalar curvature of g(0) is at least sigma(M) outside Sigma. We prove that g(0) is Einstein outside Sigma if the codimension of Sigma is at least 2. If in addition, g(0) is Lipschitz then g(0) is smooth and Einstein after a change of the smooth structure. If Sigma is a compact embedded hypersurface, g(0) is smooth up to Sigma from two sides of Sigma, and if the difference of the mean curvatures along Sigma at two sides of Sigma has a fixed appropriate sign, then g(0) is also Einstein outside Sigma. 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.