Continuous metrics and a conjecture of Schoen

TL;DR

This paper extends a classical conformal geometry theorem to singular metrics, showing such metrics are Einstein outside singularities and can be smoothed, with applications to a positive mass theorem for singular asymptotically flat manifolds.

Contribution

It generalizes the Einstein metric characterization to metrics with high-codimension singularities and proves a positive mass theorem in this singular setting.

Findings

Singular metrics achieving the Yamabe invariant are Einstein outside singularities.

Such metrics can be extended to smooth metrics on the manifold.

A positive mass theorem is established for asymptotically flat manifolds with singularities.

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.