Total curvature of convex hypersurfaces in Cartan-Hadamard manifolds

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