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PreprintCC BY-NC-SA 4.0arXiv

Codimension-three regularity of noncollapsed Ricci limit spaces under an integral volume-deficit bound

  • Lingling Kong
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Abstract

Let (X,d,p) be a noncollapsed pointed Gromov--Hausdorff limit of complete n-dimensional Riemannian manifolds with a uniform lower Ricci curvature bound, where n≥4. We assume that, on each bounded ball, the integral of the 3/2 power of the small-ball volume deficit relative to the hyperbolic comparison volume is O(r3) as r→0. We prove that the metric singular set has Hausdorff dimension at most n−3 and sigma-finite (n−3)-dimensional Hausdorff measure, thus confirming a particular case of codimension-three regularity conjecture \cite[Conjecture 2.4]{Naber2020}.

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Cite

BibTeXRIS
BibTeX
@misc{kong2026codimensionthree,
  title = {{Codimension-three regularity of noncollapsed Ricci limit spaces under an integral volume-deficit bound}},
  author = {Lingling Kong},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21452},
  eprint = {2609.21452},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21452},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Codimension-three regularity of noncollapsed Ricci limit spaces under an integral volume-deficit bound
AU  - Lingling Kong
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21452
UR  - https://arxiv.org/abs/2609.21452
AB  - Let (X,d,p) be a noncollapsed pointed Gromov--Hausdorff limit of complete n-dimensional Riemannian manifolds with a uniform lower Ricci curvature bound, where n≥4. We assume that, on each bounded ball, the integral of the 3/2 power of the small-ball volume deficit relative to the hyperbolic comparison volume is O(r^3) as r→0. We prove that the metric singular set has Hausdorff dimension at most n−3 and sigma-finite (n−3)-dimensional Hausdorff measure, thus confirming a particular case of codimension-three regularity conjecture \cite[Conjecture 2.4]{Naber2020}.
ER  -