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≥ 2. On a closed n-manifold, we show that an L^∞-Riemannian metric whose Ricci curvature is bounded below outside a singular set of codimension > 3− 1/(n−1) canonically extends to an 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^∞ singularities of codimension >3−1/3 extends smoothly across the singular set, possibly after changing the smooth structure. In higher dimensions, we construct a C^1,α-Riemannian manifold structure on the regular set of a non-collapsed RCD space that is a Riemannian manifold with bounded |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^∞ and sufficiently close to a smooth background metric.

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BibTeX RIS

BibTeX

@misc{antonelli2026removability,
  title = {{Removability of non-isolated singularities for Einstein metrics and RCD spaces}},
  author = {Gioacchino Antonelli and Gábor Székelyhidi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01464},
  eprint = {2609.01464},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01464},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Removability of non-isolated singularities for Einstein metrics and RCD spaces
AU  - Gioacchino Antonelli
AU  - Gábor Székelyhidi
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01464
UR  - https://arxiv.org/abs/2609.01464
AB  - In this paper we establish removable singularities results for Einstein metrics and for metrics with Ricci curvature bounded below. Let n≥ 2. On a closed n-manifold, we show that an L^∞-Riemannian metric whose Ricci curvature is bounded below outside a singular set of codimension > 3− 1/(n−1) canonically extends to an 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^∞ singularities of codimension >3−1/3 extends smoothly across the singular set, possibly after changing the smooth structure. In higher dimensions, we construct a C^1,α-Riemannian manifold structure on the regular set of a non-collapsed RCD space that is a Riemannian manifold with bounded |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^∞ and sufficiently close to a smooth background metric.
ER  -

Abstract

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