Curvature integral on convex hypersurfaces

Abstract

Let $M^n$, $n\ge3$, be the smooth, connected, complete, noncompact boundary of a closed convex set with nonempty interior in $\mathbb{R}^{n+1}$. We prove that its normalized scalar-curvature integral at infinity is at most $4\pi\omega_{n-2}(1-\operatorname{AVR}(M))$, except when, up to an ambient rigid motion, $M$ is the product of a smooth closed convex surface in $\mathbb{R}^3$ and $\mathbb{R}^{n-2}$. These exceptional products have zero asymptotic volume ratio and limit $8\pi\omega_{n-2}$. We also characterize the equality case for the $4\pi\omega_{n-2}(1-\operatorname{AVR}(M))$ bound. The proof uses exterior tube volumes together with the cone Steiner and Gauss--Bonnet formulas. An elementary comparison of intrinsic and Euclidean radial distances proves existence of the intrinsic limit and identifies its exact value from the cone coefficients. The same conclusions extend to orientable convex submanifolds of arbitrary codimension