Cartan-Hadamard Conjecture

may be traced back to work of André Weil in 1926

if the curvature of the ambient Cartan-Hadamard manifold M is bounded above by a nonpositive constant k, then the least perimeter enclosures in M, for any given volume, cannot have smaller perimeter than a sphere enclosing the same volume in the model space of constant curvature k

The conjecture, in all dimensions, was first stated explicitly in 1976 by Thierry Aubin.

In 2026, Shibing Chen, Mohammad Ghomi, and Peng Wang posted a proof of the conjecture in dimension 5.

In mathematics, the Cartan-Hadamard conjecture is a fundamental problem in Riemannian geometry and geometric measure theory which states that the classical isoperimetric inequality may be generalized to spaces of nonpositive sectional curvature, known as Cartan-Hadamard manifolds.

named after French mathematicians Élie Cartan and Jacques Hadamard

named after French mathematicians Élie Cartan and Jacques Hadamard