Total curvature of convex hypersurfaces in Cartan-Hadamard manifolds

Abstract

We show that if the curvature of a Cartan-Hadamard $n$-manifold is constant near a convex hypersurface $\Gamma$, then the total Gauss-Kronecker curvature $\mathcal{G}(\Gamma)$ is not less than that of any convex hypersurface nested inside $\Gamma$. This extends Borb\'{e}ly's monotonicity theorem in hyperbolic space. It follows that $\mathcal{G}(\Gamma)$ is bounded below by the volume of the unit sphere in Euclidean space $\mathbf{R}^n$.