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.
The conjecture, which is named after French mathematicians Élie Cartan and Jacques Hadamard, may be traced back to work of André Weil in 1926.
The conjecture, in all dimensions, was first stated explicitly in 1976 by Thierry Aubin,[1] and a few years later by Misha Gromov,[2][3] Yuri Burago and Viktor Zalgaller.
Mohammad Ghomi and Joel Spruck have shown that Kleiner's approach will work in all dimensions where the total curvature inequality holds.
[14] The generalized conjecture also holds for regions of small volume in all dimensions, as proved by Frank Morgan and David Johnson.