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PreprintCC BY-NC-SA 4.0arXiv

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

  • Lingling Kong
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Abstract

Let (X,d,p) be an n-dimensional noncollapsed Ricci limit space, where n≥4. Under an integral volume-deficit bound, we prove that the metric singular set has Hausdorff codimension at least four and sigma-finite (n−4)-dimensional Hausdorff measure. This establishes a special case of the codimension-four regularity conjecture. Moreover, the nonmanifold locus is closed and has locally finite (n−4)-dimensional Hausdorff measure, and its intersection with each bounded ball satisfies a tubular-volume estimate of order r4. In dimension four, the nonmanifold points form a locally finite set. A flat quotient example shows that the codimension bound is sharp.

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Cite

BibTeXRIS
BibTeX
@misc{kong2026codimensionfour,
  title = {{Codimension-four 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.2609.33995},
  eprint = {2609.33995},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33995},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Codimension-four regularity of noncollapsed Ricci limit spaces under an integral volume-deficit bound
AU  - Lingling Kong
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33995
UR  - https://arxiv.org/abs/2609.33995
AB  - Let (X,d,p) be an n-dimensional noncollapsed Ricci limit space, where n≥4. Under an integral volume-deficit bound, we prove that the metric singular set has Hausdorff codimension at least four and sigma-finite (n−4)-dimensional Hausdorff measure. This establishes a special case of the codimension-four regularity conjecture. Moreover, the nonmanifold locus is closed and has locally finite (n−4)-dimensional Hausdorff measure, and its intersection with each bounded ball satisfies a tubular-volume estimate of order r^4. In dimension four, the nonmanifold points form a locally finite set. A flat quotient example shows that the codimension bound is sharp.
ER  -