Removability of non-isolated singularities for Einstein metrics and RCD spaces

Abstract

In this paper we establish removable singularities results for Einstein metrics and for metrics with Ricci curvature bounded below. Let $n\ge 2$. On a closed $n$-manifold, we show that an $L^{\infty }$-Riemannian metric whose Ricci curvature is bounded below outside a singular set of codimension $>3-\frac{1}{n-1}$ canonically extends to an $\mathrm{RCD}$ space. As a consequence, using a new removable singularity theorem for Einstein metrics, we prove that in dimension $4$ any Einstein metric with $L^{\infty }$ singularities of codimension $>3-\frac{1}{3}$ extends smoothly across the singular set, possibly after changing the smooth structure. In higher dimensions, we construct a $C^{1,\alpha }$-Riemannian manifold structure on the regular set of a non-collapsed $\mathrm{RCD}$ space that is a Riemannian manifold with bounded $|\mathrm{Ric}|$ outside a set of codimension $>2$. Our results can be used to give a proof of Schoen's conjecture on scalar curvature singularities for metrics that are either continuous, or $L^{\infty }$ and sufficiently close to a smooth background metric.

44 pages. Comments are welcome!