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 $Γ$, then the total Gauss-Kronecker curvature $\mathcal{G}(Γ)$ is not less than that of any convex hypersurface nested inside $Γ$. This extends Borbély's monotonicity theorem in hyperbolic space. It follows that $\mathcal{G}(Γ)$ is bounded below by the volume of the unit sphere in Euclidean space $\mathbf{R}^n$.