PreprintarXiv licencearXiv
The regular set of Ricci limit spaces
Abstract
Let (X,d,p) be a nontrivial pointed Gromov--Hausdorff limit of complete connected Riemannian n-manifolds satisfying Ric≥−(n−1)g. For 1≤ j≤ n, set ℛj(X) = {x∈ X: every tangent cone at x is isometric to ℝj}, ℛ(X)=bigsqcupj=1nℛj(X). We prove that there is a unique k such that ℛ(X)=ℛk(X), answering the pointwise regular-dimension question originating in Cheeger and Colding's work~\cite[p408]{CC97}. We construct a volume-corrected heat entropy on RCD space and identify the distributional logarithmic-density defect tr(Hess f)−Δf by Green-volume regularization.
Related
- On the tangent flows of noncollapsed Ricci flow limit spaces
- Codimension-three regularity of noncollapsed Ricci limit spaces under an integral volume-deficit bound
- Codimension-four regularity of noncollapsed Ricci limit spaces under an integral volume-deficit bound
- Quantitative Volume Estimate on Singular Sets of One-phase Free Boundary Problems in Ricci Limit Spaces
- Quantitative Rigidity of Ricci Nonnegative Manifolds via the Regular Green Constant
BibTeX
@misc{ren2026regular,
title = {{The regular set of Ricci limit spaces}},
author = {Tianyin Ren},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41449},
eprint = {2610.41449},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41449},
}
RIS
TY - UNPB
TI - The regular set of Ricci limit spaces
AU - Tianyin Ren
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41449
UR - https://arxiv.org/abs/2610.41449
AB - Let (X,d,p) be a nontrivial pointed Gromov--Hausdorff limit of complete connected Riemannian n-manifolds satisfying Ric≥−(n−1)g. For 1≤ j≤ n, set ℛ_j(X) = {x∈ X: every tangent cone at x is isometric to ℝ^j}, ℛ(X)=bigsqcup_j=1^nℛ_j(X). We prove that there is a unique k such that ℛ(X)=ℛ_k(X), answering the pointwise regular-dimension question originating in Cheeger and Colding's work~\cite[p408]{CC97}. We construct a volume-corrected heat entropy on RCD space and identify the distributional logarithmic-density defect tr(Hess f)−Δf by Green-volume regularization.
ER -