Total curvature of convex hypersurfaces in Cartan-Hadamard manifolds
Mohammad Ghomi, John Ioannis Stavroulakis
TL;DR
The paper establishes a monotonicity result for the total Gauss-Kronecker curvature of convex hypersurfaces in Cartan-Hadamard manifolds: if a convex hypersurface $\Gamma$ contains a nested convex hypersurface $\gamma$ and the sectional curvature is constant on a neighborhood of $\Gamma$, then the total curvature satisfies $\mathcal{G}(\Gamma)\ge \mathcal{G}(\gamma)$. The core method is a distance-based comparison formula for total curvature, using a convex interpolation $u$ between $\gamma$ and $\Gamma$; when curvature near $\Gamma$ is constant, mixed curvature terms vanish and monotonicity follows. In dimension $n=3$, constancy on $\Gamma$ alone suffices, while for $n\ge 3$ constancy in a neighborhood is generally needed. As a consequence, the total curvature of $\Gamma$ is bounded below by the Euclidean unit-sphere measure $|\mathbf{S}^{n-1}|$, contributing to the isoperimetric-type questions in nonpositive curvature settings and extending Borbély’s hyperbolic-space result to 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 $\mathbf{R}^n$.