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Lipschitz volume rigidity for non-collapsed RCD spaces with boundary
Abstract
We study Lipschitz-volume rigidity for non-collapsed RCD(K,N) spaces, possibly with boundary. We prove that a measure-preserving $1$-Lipschitz map between such spaces restricts to an isometry from the interior of the domain onto its image, where the latter is equipped with the intrinsic length distance. We then characterize when this interior rigidity extends to an isometric bijection between the whole spaces, in terms of the behavior of the map on the boundaries. Along the way, we establish a rigidity result of independent interest in a more general class of rectifiable metric spaces.
Related
- Curvature integral, volume ratio, bi-Lipschitz for spaces with curvature bounded below
- Rigidity of volume-minimizing free boundary hypersurfaces in high dimensional Riemannian manifolds
- Quantitative Volume Estimate on Singular Sets of One-phase Free Boundary Problems in Ricci Limit Spaces
- Rigidity of Lipschitz map using harmonic map heat flow
- Length distortion of volume-preserving Lipschitz mappings
BibTeX
@misc{li2026lipschitz,
title = {{Lipschitz volume rigidity for non-collapsed $\rm RCD$ spaces with boundary}},
author = {Zhenhao Li and Enrico Pasqualetto and Xin Qian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39493},
eprint = {2610.39493},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39493},
}
RIS
TY - UNPB
TI - Lipschitz volume rigidity for non-collapsed RCD spaces with boundary
AU - Zhenhao Li
AU - Enrico Pasqualetto
AU - Xin Qian
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39493
UR - https://arxiv.org/abs/2610.39493
AB - We study Lipschitz-volume rigidity for non-collapsed RCD(K,N) spaces, possibly with boundary. We prove that a measure-preserving $1$-Lipschitz map between such spaces restricts to an isometry from the interior of the domain onto its image, where the latter is equipped with the intrinsic length distance. We then characterize when this interior rigidity extends to an isometric bijection between the whole spaces, in terms of the behavior of the map on the boundaries. Along the way, we establish a rigidity result of independent interest in a more general class of rectifiable metric spaces.
ER -