OP-00669 Solution claimed

Euclidean isoperimetry in higher-dimensional Cartan–Hadamard manifolds

Let $(M^n, g)$ be a complete simply connected smooth Riemannian manifold with $n \ge 5$ and sectional curvature everywhere nonpositive (a Cartan–Hadamard manifold). For a domain $\Omega \Subset M$ with smooth boundary write $V = \operatorname{Vol}_g(\Omega)$ and $A = \operatorname{Area}_g(\partial\Omega)$, and let $\omega_n$ be the volume of the Euclidean unit ball. Euclidean balls satisfy $A = n\omega_n^{1/n} V^{(n-1)/n}$. The same inequality holds in dimensions $n \le 4$: Weil proved it for $n = 2$, Kleiner for $n = 3$ and Croke for $n = 4$. Ghomi and Stavroulakis proved that small $C^2$ perturbations of the Euclidean metric on a ball to metrics of nonpositive curvature do not decrease the isoperimetric ratio $A^n/V^{n-1}$. Ghomi proved the inequality for Cartan–Hadamard manifolds with nullity index at least $n - 3$, and for $n = 5$ under sufficiently pinched negative curvature.

Problem. Does $$A \ge n\omega_n^{1/n} V^{(n-1)/n}$$ hold for every such $M$ and every such domain $\Omega$? No pinching, symmetry, small-volume or curvature-nullity hypothesis is imposed.

Source: The Cartan–Hadamard conjecture, after Aubin (1975), Gromov (1981) and Burago and Zalgaller (1980), as stated with its equality case by Ghomi and Stavroulakis (2026), §1; stated for n ≥ 5, without the equality case, by AIM — Open Applied Problems.

Imported from github.com/MColbrook/AIM.

Classification

Primary: DG — Differential Geometry
Secondary: AP, MG
Keywords: AIM Open Applied Problems, Cartan–Hadamard manifolds, isoperimetric inequality, nonpositive curvature

References

@article{aubin1975,
  author  = {Aubin, Thierry},
  title   = {Probl{\`e}mes isop{\'e}rim{\'e}triques et espaces de {S}obolev},
  journal = {C. R. Acad. Sci. Paris S{\'e}r. A-B},
  volume  = {280},
  number  = {5},
  year    = {1975},
  pages   = {A279--A281},
}
@book{gromov1981,
  author    = {Gromov, Misha},
  title     = {Structures m{\'e}triques pour les vari{\'e}t{\'e}s riemanniennes},
  editor    = {Lafontaine, J. and Pansu, P.},
  series    = {Textes Math{\'e}matiques},
  volume    = {1},
  publisher = {CEDIC},
  address   = {Paris},
  year      = {1981},
}
@book{burago1980,
  author    = {Burago, Yu. D. and Zalgaller, V. A.},
  title     = {Geometricheskie neravenstva},
  publisher = {Nauka, Leningrad. Otdel.},
  address   = {Leningrad},
  year      = {1980},
}
@article{weil1926,
  author  = {Weil, Andr{\'e}},
  title   = {Sur les surfaces {\`a} courbure n{\'e}gative},
  journal = {C. R. Acad. Sci. Paris},
  volume  = {182},
  year    = {1926},
  pages   = {1069--1071},
}
@article{kleiner1992,
  author  = {Kleiner, Bruce},
  title   = {An isoperimetric comparison theorem},
  journal = {Invent. Math.},
  volume  = {108},
  number  = {1},
  year    = {1992},
  pages   = {37--47},
  doi     = {10.1007/BF02100598},
}
@article{croke1984,
  author  = {Croke, Christopher B.},
  title   = {A sharp four dimensional isoperimetric inequality},
  journal = {Comment. Math. Helv.},
  volume  = {59},
  number  = {1},
  year    = {1984},
  pages   = {187--192},
  doi     = {10.1007/BF02566344},
}
@article{ghomi2026local,
  author        = {Ghomi, Mohammad and Stavroulakis, John Ioannis},
  title         = {A local isoperimetric inequality for balls with nonpositive curvature},
  journal       = {Int. Math. Res. Not. IMRN},
  volume        = {2026},
  number        = {6},
  year          = {2026},
  pages         = {rnag043},
  doi           = {10.1093/imrn/rnag043},
  eprint        = {2505.11971},
  archivePrefix = {arXiv},
}
@misc{ghomi2026nullity,
  author        = {Ghomi, Mohammad},
  title         = {Isoperimetric and total curvature inequalities in {C}artan--{H}adamard manifolds with nullity},
  year          = {2026},
  eprint        = {2605.24638},
  archivePrefix = {arXiv},
}
@misc{ghomi2026pinched,
  author        = {Ghomi, Mohammad},
  title         = {Total curvature and isoperimetric inequalities in pinched {C}artan--{H}adamard manifolds},
  year          = {2026},
  eprint        = {2608.12020},
  archivePrefix = {arXiv},
}
@misc{aim-cartan-hadamard-isoperimetry,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Euclidean isoperimetry in higher-dimensional Cartan–Hadamard manifolds}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/127-cartan-hadamard-isoperimetry.md},
  note         = {Entry 127 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Generalized Cartan--Hadamard isoperimetry and Euclidean equality rigidity v1 GitHub ↗

OpenAI

We resolve the generalized Cartan--Hadamard isoperimetric conjecture in every dimension. In a complete simply connected smooth manifold with sectional curvature at most $\kappa\le0$, every finite-volume set of finite ambient perimeter has perimeter at least that of the equal-volume ball in curvature $\kappa$. For bounded positive-volume sets, equality in the Euclidean comparison holds precisely when the set agrees up to null …

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: math @ 1581d4c, build/main.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00669, "Euclidean isoperimetry in higher-dimensional Cartan–Hadamard manifolds", clopen.solutions.