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Quantitative Centre Rigidity under Riemannian Approximation in Heisenberg Groups
Abstract
For n≥2, we prove a quantitative centre-rigidity estimate for bounded sets of finite Riemannian perimeter that are stationary and volume-constrained stable for normalized Riemannian perimeter with prescribed curvature hj. Under confinement to a fixed compact set and cutoffs of vanishing full surface Dirichlet energy on the smooth regular boundary, the squared L2 norms of the horizontal centre derivative, angle defect and weighted shape defect are bounded by C(εj2+|hj−2n|C2)|Ωj| when the parenthesized quantity is sufficiently small. The excess of normalized Riemannian perimeter over horizontal perimeter satisfies the same bound.
Related
- The isoperimetric problem in the Riemannian Heisenberg $3$-space
- Spectral asymptotics of sub-Riemannian Laplacians on compact Heisenberg manifolds
- Stable Rigidity in the Heisenberg Groups
- Rigidity of volume-minimizing free boundary hypersurfaces in high dimensional Riemannian manifolds
- On the Geometry of Wasserstein Barycenter II: Riemannian Rigidity, Essential Non-Branching, and Finsler Models
BibTeX
@misc{chen2026quantitative,
title = {{Quantitative Centre Rigidity under Riemannian Approximation in Heisenberg Groups}},
author = {Wei-Chun Chen and Wei-Ting Kao and Chun-Kai Lien},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03904},
eprint = {2610.03904},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03904},
}
RIS
TY - UNPB
TI - Quantitative Centre Rigidity under Riemannian Approximation in Heisenberg Groups
AU - Wei-Chun Chen
AU - Wei-Ting Kao
AU - Chun-Kai Lien
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03904
UR - https://arxiv.org/abs/2610.03904
AB - For n≥2, we prove a quantitative centre-rigidity estimate for bounded sets of finite Riemannian perimeter that are stationary and volume-constrained stable for normalized Riemannian perimeter with prescribed curvature h_j. Under confinement to a fixed compact set and cutoffs of vanishing full surface Dirichlet energy on the smooth regular boundary, the squared L^2 norms of the horizontal centre derivative, angle defect and weighted shape defect are bounded by C(ε_j^2+|h_j−2n|_C^2)|Ω_j| when the parenthesized quantity is sufficiently small. The excess of normalized Riemannian perimeter over horizontal perimeter satisfies the same bound.
ER -