Total curvature of convex hypersurfaces in Cartan-Hadamard manifolds

TL;DR

This paper extends a monotonicity theorem for convex hypersurfaces in Cartan-Hadamard manifolds, showing that under certain curvature conditions, the total Gauss-Kronecker curvature is bounded below by the Euclidean sphere volume.

Contribution

It generalizes Borbély's theorem from hyperbolic space to Cartan-Hadamard manifolds with constant curvature near the hypersurface.

Findings

Total curvature is bounded below by Euclidean sphere volume.

Monotonicity property extends to more general manifolds.

Curvature conditions influence hypersurface geometry.

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 $\mathbf{R}^n$.