Continuous metrics and a conjecture of Schoen

Abstract

A classical theorem in conformal geometry states that on a manifold with non-positive Yamabe invariant, a smooth metric achieving the invariant must be Einstein. In this work, we extend it to the singular case and show that in all dimension, if a continuous metric is smooth outside a compact set of high co-dimension and achieves the Yamabe invariant, then the metric is Einstein away from the singularity and can be extended to be smooth on the manifold in a suitable sense. As an application of the method, we prove a Positive Mass Theorem for asymptotically flat manifolds with analogous singularities.

minor revision, printing mistakes corrected

References in corpus (4)

• Scalar curvature and singular metrics
• Positive scalar curvature with skeleton singularities
• Pointwise lower scalar curvature bounds for $C^0$ metrics via regularizing Ricci flow
• Weak scalar curvature lower bounds along Ricci flow