Minimal filling volume of a simple manifold
Let $n \ge 2$. A compact Riemannian manifold $(M^n, g)$ is simple if its boundary is strictly convex and any two points are joined by a unique minimizing geodesic depending smoothly on its endpoints; a simple manifold is diffeomorphic to a ball. It is a minimal filling if every compact oriented Riemannian $n$-manifold $(N, h)$ with $\partial N = \partial M$ whose intrinsic distance satisfies $$d_h(x, y) \ge d_g(x, y) \qquad (x, y \in \partial M)$$ has $\operatorname{Vol}_h(N) \ge \operatorname{Vol}_g(M)$; the topology of $N$ is unrestricted. Burago and Ivanov proved that a metric on a region of $\mathbb{R}^n$ with connected boundary that is sufficiently $C^2$-close to the Euclidean metric is a minimal filling, and proved the same for metrics close to a hyperbolic one; Ruan extended this to metrics close to a negatively curved symmetric metric. In dimension two Ivanov proved that every simple surface is a minimal filling among fillings homeomorphic to the disc. Since the hemisphere is a limit of simple manifolds, an affirmative answer would imply Gromov's filling area conjecture, that every orientable surface isometrically filling the circle of length $2\pi$ has area at least $2\pi$ (Ivanov).
Problem. Is every simple Riemannian manifold of dimension $n \ge 2$ a minimal filling? No statement about the equality case is asked for.
Source: Ivanov (2010), Conjecture 1.6, in his ICM 2010 lecture.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, boundary distance functions, minimal fillings
References
@inproceedings{ivanov2010,
author = {Ivanov, Sergei},
title = {Volume comparison via boundary distances},
booktitle = {Proceedings of the International Congress of Mathematicians 2010 (ICM 2010), Vol. II},
publisher = {Hindustan Book Agency},
address = {New Delhi},
year = {2010},
pages = {769--784},
doi = {10.1142/9789814324359_0073},
eprint = {1004.2505},
archivePrefix = {arXiv},
}
@article{burago2010,
author = {Burago, Dmitri and Ivanov, Sergei},
title = {Boundary rigidity and filling volume minimality of metrics close to a flat one},
journal = {Ann. of Math. (2)},
volume = {171},
number = {2},
year = {2010},
pages = {1183--1211},
doi = {10.4007/annals.2010.171.1183},
}
@article{burago2013,
author = {Burago, Dmitri and Ivanov, Sergei},
title = {Area minimizers and boundary rigidity of almost hyperbolic metrics},
journal = {Duke Math. J.},
volume = {162},
number = {7},
year = {2013},
pages = {1205--1248},
doi = {10.1215/00127094-2142529},
eprint = {1011.1570},
archivePrefix = {arXiv},
}
@article{ruan2025,
author = {Ruan, Yuping},
title = {Filling volume minimality and boundary rigidity of metrics close to a negatively curved symmetric metric},
journal = {Math. Ann.},
volume = {391},
number = {1},
year = {2025},
pages = {1065--1122},
doi = {10.1007/s00208-024-02941-6},
eprint = {2201.09175},
archivePrefix = {arXiv},
}
@article{ivanov2011,
author = {Ivanov, Sergei},
title = {Filling minimality of {F}inslerian 2-discs},
journal = {Proc. Steklov Inst. Math.},
volume = {273},
number = {1},
year = {2011},
pages = {176--190},
doi = {10.1134/S0081543811040079},
eprint = {0910.2257},
archivePrefix = {arXiv},
}
@article{gromov1983,
author = {Gromov, Mikhael},
title = {Filling {R}iemannian manifolds},
journal = {J. Differential Geom.},
volume = {18},
number = {1},
year = {1983},
pages = {1--147},
doi = {10.4310/jdg/1214509283},
}
@misc{aim-simple-manifold-minimal-filling,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Minimal filling volume of a simple manifold}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/121-simple-manifold-minimal-filling.md},
note = {Entry 121 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-00557, "Minimal filling volume of a simple manifold", clopen.solutions.