The volume of a slightly curved submanifold in a convex region

Abstract:

Let T be a compact convex region in an n-dimensional Riemannian space, ${k_s}$ be the minimum sectional curvature in T, and $\kappa > 0$ be the minimum normal curvature of the boundary of T. Denote by ${P^\nu }(\xi )$ a v-dimensional sphere, plane or hyperbolic plane of curvature $\xi$. We assume that ${k_s}$, k are such that on ${P^2}({k_s})$ there exists a circumference of curvature k. Let ${R_0} = {R_0}(\kappa ,{k_s})$ be its radius. Now, let Q be a convex (in interior sense) m-dimensional surface in T whose normal curvatures with respect to any normal are not greater than x satisfying $0 \leqslant \chi < \kappa$. Denote by ${L_\chi }$ the length of a circular arc of curvature x in ${P^2}({k_s})$ with the distance $2{R_0}$ between its ends. We prove that the volume of Q does not exceed the volume of a ball in ${P^m}({k_s} - (n - m){\chi ^2})$ of radius $\tfrac {1}{2}{L_\chi }$. These volumes are equal when T is a ball in ${P^n}({k_s})$ and Q is its m-dimensional diameter.