A geometric rigidity estimate for codimension-1 immersions into spheres
Abstract
We establish a quantitative stability estimate for codimension-one immersions into a round sphere. Let (M,g) be an oriented Riemannian manifold and suppose that a prescribed shape operator is realised by a smooth isometric immersion θ:(M,g)→𝕊n+1. We prove that, on every relatively compact strongly Lipschitz domain and for every 1<p<∞, any Sobolev immersion ϕ is close to θ modulo an ambient rotation, with the W1,p-distances between both the immersions and their Gauss maps controlled by the Lp stretching-plus-bending energies of ϕ. No {\it a priori} bounds on the fundamental forms of ϕ are required. The proof proceeds by extending the immersions along normal geodesics and reducing the problem to an equidimensional geometric rigidity estimate on the sphere. A finite localisation and patching argument handles the possible non-injectivity of the normal extension of the reference immersion.
Related
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- Rigidity of Ricci-Pinched Complete Self-Shrinkers in Arbitrary Codimension
BibTeX
@misc{li2026geometric,
title = {{A geometric rigidity estimate for codimension-1 immersions into spheres}},
author = {Siran Li and Isaac Newell},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.37665},
eprint = {2609.37665},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37665},
}
RIS
TY - UNPB
TI - A geometric rigidity estimate for codimension-1 immersions into spheres
AU - Siran Li
AU - Isaac Newell
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2609.37665
UR - https://arxiv.org/abs/2609.37665
AB - We establish a quantitative stability estimate for codimension-one immersions into a round sphere. Let (M,g) be an oriented Riemannian manifold and suppose that a prescribed shape operator is realised by a smooth isometric immersion θ:(M,g)→𝕊^n+1. We prove that, on every relatively compact strongly Lipschitz domain and for every 1<p<∞, any Sobolev immersion ϕ is close to θ modulo an ambient rotation, with the W^1,p-distances between both the immersions and their Gauss maps controlled by the L^p stretching-plus-bending energies of ϕ. No {\it a priori} bounds on the fundamental forms of ϕ are required. The proof proceeds by extending the immersions along normal geodesics and reducing the problem to an equidimensional geometric rigidity estimate on the sphere. A finite localisation and patching argument handles the possible non-injectivity of the normal extension of the reference immersion.
ER -