A strictly convex function on every simple manifold
Let $(M, g)$ be a smooth compact Riemannian manifold with boundary, of dimension $n \ge 3$, and assume it is simple: its boundary is strictly convex and any two points are joined by a unique geodesic, depending smoothly on its endpoints. For $\rho \in C^\infty(M; \mathbb{R})$ write $\operatorname{Hess}_g \rho(v, v) = g(\nabla_v \nabla \rho, v)$. Paternain, Salo, Uhlmann and Zhou showed that a compact manifold with strictly convex boundary carries a smooth function with positive definite Hessian if it is simply connected with nonpositive sectional curvature, or simply connected without focal points, or has nonnegative sectional curvature, and that a compact surface with strictly convex boundary carries one exactly when it is non-trapping, building on Betelú, Gulliver and Littman; in particular every simple surface does. Stefanov, Uhlmann and Vasy proved that, in dimension $n \ge 3$, a compact manifold with strictly convex boundary carrying such a function with $\partial M$ as a level set is lens rigid.
Problem. Must there exist $\rho \in C^\infty(M; \mathbb{R})$ and a constant $c > 0$ such that $$\operatorname{Hess}_g \rho(x)(v, v) \ge c\, g_x(v, v) \qquad \text{for every } x \in M,\ v \in T_xM?$$ No condition that $\rho$ be constant on the boundary is imposed.
Source: Asked by Ilmavirta and Monard (2019), §9, Open question 9 (whether simple manifolds of dimension at least three admit strictly convex foliations); noted as open by Paternain, Salo and Uhlmann (2023), §15.2, and by Ilmavirta, Saksala and Yan (2026), introduction; 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, boundary rigidity, simple manifolds, strictly convex functions
References
@book{paternain2023,
author = {Paternain, Gabriel P. and Salo, Mikko and Uhlmann, Gunther},
title = {Geometric Inverse Problems: With Emphasis on Two Dimensions},
series = {Cambridge Studies in Advanced Mathematics},
volume = {204},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {2023},
doi = {10.1017/9781009039901},
note = {\S 15.2},
}
@incollection{ilmavirta2019,
author = {Ilmavirta, Joonas and Monard, Fran{\c{c}}ois},
title = {Integral geometry on manifolds with boundary and applications},
booktitle = {The Radon Transform: The First 100 Years and Beyond},
editor = {Ramlau, Ronny and Scherzer, Otmar},
series = {Radon Ser. Comput. Appl. Math.},
volume = {22},
publisher = {De Gruyter},
address = {Berlin},
year = {2019},
pages = {43--114},
doi = {10.1515/9783110560855-004},
eprint = {1806.06088},
archivePrefix = {arXiv},
note = {\S 9, Open question 9},
}
@misc{ilmavirta2026,
author = {Ilmavirta, Joonas and Saksala, Teemu and Yan, Lili},
title = {Rigidity of homogeneous {L}am{\'e} systems},
year = {2026},
eprint = {2602.08860},
archivePrefix = {arXiv},
}
@article{paternain2019,
author = {Paternain, Gabriel P. and Salo, Mikko and Uhlmann, Gunther and Zhou, Hanming},
title = {The geodesic {X}-ray transform with matrix weights},
journal = {Amer. J. Math.},
volume = {141},
number = {6},
year = {2019},
pages = {1707--1750},
doi = {10.1353/ajm.2019.0045},
eprint = {1605.07894},
archivePrefix = {arXiv},
}
@article{betelu2002,
author = {Betel{\'u}, Santiago and Gulliver, Robert and Littman, Walter},
title = {Boundary control of {PDE}s via curvature flows: the view from the boundary, {II}},
journal = {Appl. Math. Optim.},
volume = {46},
number = {2},
year = {2002},
pages = {167--178},
doi = {10.1007/s00245-002-0742-6},
}
@article{stefanov2021,
author = {Stefanov, Plamen and Uhlmann, Gunther and Vasy, Andr{\'a}s},
title = {Local and global boundary rigidity and the geodesic {X}-ray transform in the normal gauge},
journal = {Ann. of Math. (2)},
volume = {194},
number = {1},
year = {2021},
pages = {1--95},
doi = {10.4007/annals.2021.194.1.1},
eprint = {1702.03638},
archivePrefix = {arXiv},
}
@misc{aim-convex-function-on-simple-manifold,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A strictly convex function on every simple manifold}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/069-convex-function-on-simple-manifold.md},
note = {Entry 069 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00512, "A strictly convex function on every simple manifold", clopen.solutions.