Total curvature and the isoperimetric inequality in Cartan-Hadamard manifolds

Mohammad Ghomi, Joel Spruck

Published 2019-08-26Version 1

We prove that the total positive Gauss-Kronecker curvature of any closed hypersurface embedded in a complete simply connected manifold of nonpositive curvature $M^n$, $n\geq 2$, is bounded below by the volume of the unit sphere in Euclidean space $\mathbf{R}^n$. This yields the optimal isoperimetric inequality for bounded regions of finite perimeter in $M$, via Kleiner's variational approach, and thus settles the Cartan-Hadamard conjecture. The proof employs a comparison formula for total curvature of level sets in Riemannian manifolds, and estimates for smooth approximation of the signed distance function. Immediate applications include sharp extensions of the Faber-Krahn and Sobolev inequalities to manifolds of nonpositive curvature.