Generalized Cartan–Hadamard conjecture (Aubin conjecture)

For every integer $n\ge 2$ , every complete simply connected Riemannian $n$ -manifold $(M^n,g)$ with sectional curvature $\operatorname{Sec}_g\le \bar k\le 0$ , and every volume $V>0$ , the isoperimetric profile satisfies $I_g(V)\ge I_{\bar k}(V)$ , where $I_g(V):=\inf\{\operatorname{Per}_g(\Omega):\operatorname{Vol}_g(\Omega)=V\}$ and $I_{\bar k}$ is the isoperimetric profile of the simply connected space form of constant sectional curvature $\bar k$ . Equivalently, among regions of fixed volume, the corresponding model-space geodesic balls have no greater perimeter than any region in $(M^n,g)$ .

References

Primary source

arXiv

Additional references

A proof of the Cartan-Hadamard conjecture for small volumes under a Ricci curvature lower bound — arXiv — Marcos Agnoletto, Márcio Fabiano Da Silva, Stefano Nardulli, Reinaldo Resende

differential-geometry geometric-analysis isoperimetric-inequalities

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed progress

A new preprint proves the conjectured comparison for sufficiently small volumes in every dimension, while the full arbitrary-volume conjecture remains open despite additional restricted claims.

The generalized Cartan–Hadamard conjecture asserts that, under nonpositive curvature, spheres in the constant-curvature model minimize perimeter among regions of fixed volume. The arbitrary-volume statement is not settled in general.

Known results

Dimension $2$ : established by Gerrit Bol.
Dimension $3$ : established by Bruce Kleiner.
Further higher-dimensional results hold only under additional curvature or geometric hypotheses.

September 2026 developments

Agnoletto, Da Silva, Nardulli, and Resende claim the comparison for sufficiently small volumes in every dimension $n\ge 2$ and reduce the arbitrary-volume case to rigidity of equality; this is a substantial partial result, not a full solution.
Chen, Ghomi, and Wang claim a complete proof in dimension $5$ .
Wheeler claims an all-dimensional proof under an additional curvature-pinching condition. These preprint claims are unverified.

Current status (as of September 2026): The small-volume regime and several restricted settings are claimed, but the general arbitrary-volume conjecture remains open and the new claims have not been independently verified.

Sources

RemarkAI-assistedClaimed by OpenAI. Claims generalized Cartan–Hadamard isoperimetry in all dimensions for complete simply connected smooth manifolds with sectional curvature bounded above by a nonpositive constant, and Euclidean equality rigidity for bounded positive-volume sets.See full solutionHide full solution

Claimed by OpenAI. Claims generalized Cartan–Hadamard isoperimetry in all dimensions for complete simply connected smooth manifolds with sectional curvature bounded above by a nonpositive constant, and Euclidean equality rigidity for bounded positive-volume sets.

Scope relative to this problem: The source claims the generalized smooth Cartan-Hadamard isoperimetric comparison for complete simply connected manifolds in all dimensions with sectional curvature bounded above by a nonpositive constant. Its Euclidean equality rigidity is for bounded positive-volume sets. Retain the source smooth setting and stated set/perimeter conventions, rather than inferring unrestricted singular metric-space rigidity.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026/paper.pdf

OpenAI-337-01-Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity.pdf540,830 bytesOpen