The Cartan-Hadamard isoperimetric inequality under controlled variation of curvature

Abstract

We prove the Cartan-Hadamard conjecture (the sharp Euclidean isoperimetric inequality) in every dimension, assuming that the sectional curvature is pinched relative to a reference radius of curvature. The reference radius of curvature may vary, and the condition includes examples where the radius of curvature blows up, and the manifold has both unbounded (negative) and asymptotically vanishing sectional curvature. The key step in the proof is to control the entire convex hull of a quotient minimiser at its mean-curvature scale and compare the metric distortion with the surplus in the normalised hyperbolic isoperimetric profile at that same scale.