Continuous metrics and a conjecture of Schoen

Source abstract

A classical theorem in conformal geometry states that on a compact manifold with nonpositive Yamabe invariant σ 0 \sigma _0 , a smooth metric achieving the invariant must be Einstein. In this work, we extend it to the singular case and show that in all dimensions, if a continuous metric is smooth outside a compact set Σ \Sigma of co-dimension 2 + a 2+a for some a > 0 a>0 with unit volume and with scalar curvature bounded below by σ 0 \sigma _0 , then the metric is Einstein away from Σ \Sigma and is isometric to a smooth Einstein manifold of scalar curvature σ 0 \sigma _0 . In this sense, singular metric can be extended to be smooth on the manifold in a suitable sense. This is related to a conjecture of Schoen in the continuous setting. As an application of the method, we prove a positive mass theorem for asymptotically flat manifolds with analogous singularities. This is based on a new maximum principle on Ricci-DeTurck flows which allow initial data to be singular.