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.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2021
- DOI:
- 10.48550/arXiv.2111.05582
- arXiv:
- arXiv:2111.05582
- Bibcode:
- 2021arXiv211105582L
- Keywords:
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- Mathematics - Differential Geometry
- E-Print:
- minor revision, printing mistakes corrected