Rigidity of positively curved shells above one-half Hölder regularity
Let $g$ be a smooth Riemannian metric on $S^2$ with positive Gauss curvature. An isometric embedding of $(S^2, g)$ of class $C^{1,\alpha}$ is an embedding $u : S^2 \to \mathbb{R}^3$ with locally $\alpha$-Hölder continuous first derivatives and $Du(x)^T Du(x) = g(x)$ in local coordinates; an embedding is an injective immersion that is a homeomorphism onto its image.
For $C^2$ isometric immersions the image is convex and unique up to rigid motions, while by the Nash–Kuiper theorem $C^1$ isometric embeddings are flexible. Borisov proved that for $\alpha > 2/3$ the image of every $C^{1,\alpha}$ isometric immersion of $(S^2, g)$ is the boundary of a bounded convex set (Conti, De Lellis and Székelyhidi gave a short proof). In the other direction, isometric immersions of two-dimensional discs satisfy an h-principle in $C^{1,\alpha}$ for $\alpha < 1/5$ (De Lellis, Inauen and Székelyhidi), and a 2025 preprint of Cao, Hirsch and Inauen extends this to $\alpha < 1/3$. De Lellis and Inauen proved that for $\alpha > 1/2$ the Levi-Civita connection of every $C^{1,\alpha}$ isometric immersion is induced by the Euclidean connection, and constructed $C^{1,\alpha}$ isometric embeddings of portions of the round sphere for which this fails when $\alpha < 1/2$; Cao and Inauen proved the analogous dichotomy at $1/2$ for isometric extensions.
Problem. For every such metric $g$, every $\alpha \in (1/2, 1)$ and every $C^{1,\alpha}$ isometric embedding $u$ of $(S^2, g)$ into $\mathbb{R}^3$, must $u(S^2)$ be the boundary of a convex body?
Source: Conti, De Lellis and Székelyhidi (2012), §1.3, who suggest 1/2 as the likely critical exponent, after Borisov (2004); see also De Lellis (2020 lecture), where the threshold 1/2 is described as expected. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, convex integration, convex surfaces, isometric embeddings
References
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author = {De Lellis, Camillo},
title = {Rigidity and flexibility of isometric embeddings},
howpublished = {Mathematics Colloquium, University of California San Diego, 6 February 2020, lecture abstract},
year = {2020},
url = {https://math.ucsd.edu/seminar/rigidity-and-flexibility-isometric-embeddings},
}
@incollection{conti2012,
author = {Conti, Sergio and De Lellis, Camillo and Sz{\'e}kelyhidi, Jr., L{\'a}szl{\'o}},
title = {h-principle and rigidity for {$C^{1,\alpha}$} isometric embeddings},
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eprint = {0905.0370},
archivePrefix = {arXiv},
}
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}
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author = {Borisov, Yu. F.},
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}
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author = {Borisov, Yu. F.},
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}
}
@article{delellis2018,
author = {De Lellis, Camillo and Inauen, Dominik and Sz{\'e}kelyhidi, Jr., L{\'a}szl{\'o}},
title = {A {N}ash--{K}uiper theorem for {$C^{1,1/5-\delta}$} immersions of surfaces in 3 dimensions},
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eprint = {1510.01934},
archivePrefix = {arXiv},
}
@misc{cao2025,
author = {Cao, Wentao and Hirsch, Jonas and Inauen, Dominik},
title = {A {N}ash--{K}uiper theorem for isometric immersions beyond {B}orisov's exponent},
year = {2025},
eprint = {2503.13867},
archivePrefix = {arXiv},
}
@article{delellis2020caps,
author = {De Lellis, Camillo and Inauen, Dominik},
title = {{$C^{1,\alpha}$} isometric embeddings of polar caps},
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eprint = {1809.04161},
archivePrefix = {arXiv},
}
@article{cao2024,
author = {Cao, Wentao and Inauen, Dominik},
title = {Rigidity and flexibility of isometric extensions},
journal = {Comment. Math. Helv.},
volume = {99},
number = {1},
year = {2024},
pages = {39--80},
doi = {10.4171/CMH/564},
eprint = {2010.00418},
archivePrefix = {arXiv},
}
@misc{aim-holder-isometric-shell-rigidity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Rigidity of positively curved shells above one-half Hölder regularity}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/235-holder-isometric-shell-rigidity.md},
note = {Entry 235 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00598, "Rigidity of positively curved shells above one-half Hölder regularity", clopen.solutions.