Cartan–Hadamard isoperimetric conjecture

The Cartan–Hadamard isoperimetric conjecture asserts that Euclidean space is extremal for the isoperimetric problem among nonpositively curved spaces: in a complete simply connected Riemannian $n$-manifold with sectional curvature $K\le0$, every bounded smooth domain $\Omega$ should satisfy the sharp Euclidean inequality $\operatorname{Area}(\partial\Omega)\ge n\,\omega_n^{1/n}\operatorname{Vol}(\Omega)^{(n-1)/n}$, in which equality holds for round balls in $\mathbb R^n$. The first explicit all-dimensional formulation identified in the literature is Aubin's, from his 1976 work on isoperimetric problems and Sobolev spaces [Aubin1976NonlinearProblems].

The two-dimensional case is classical. In higher dimensions the landmark results are Kleiner's isoperimetric comparison theorem in dimension three [Kleiner1992Isoperimetric] and Croke's sharp four-dimensional inequality [Croke1984SharpFourDimensional], so the conjecture holds for all $n\le4$; special geometries in higher dimensions are also covered.

For every $n\ge5$ the unrestricted conjecture is open. The difficulty is not an inequality of the right shape but the sharp Euclidean constant itself, and a resolution would require techniques going beyond the methods that succeeded in dimensions three and four.