Inequalities of Gauss-Bonnet type for a convex domain

Abstract

Let N N be a compact convex n n -dimensional Riemannian manifold with a boundary ∂ N \partial N having normal curvatures ⩾ κ > 0 \geqslant \kappa > 0 . Suppose the sectional curvature > − κ 2 > - {\kappa ^2} in N N . Let H H be the integral mean curvature of ∂ N \partial N , V V be the volume of N N , k s c {k_{sc}} be the scalar curvature and k ¯ R ( p ) {\bar k_R}(p) , p ∈ N p \in N , be the maximum Ricci curvature at p p . Then \[ H ⩾ n − 2 2 κ 2 V − 1 2 ( n − 1 ) ∫ N k s c d V , H ⩾ ( n − 2 ) κ 2 V − 1 n − 1 ∫ N k ¯ R d V . H \geqslant \frac {{n - 2}} {2}{\kappa ^2}V - \frac {1} {{2(n - 1)}}\int _N {{k_{sc}}\;dV} ,\quad H \geqslant (n - 2){\kappa ^2}V - \frac {1} {{n - 1}}\int _N {{{\bar k}_R}\;dV.} \] Let N − {N_ - } denote N N with nonpositive sectional curvature. Let G G be the integral Gauss curvature of ∂ N − \partial {N_ - } . Then G ⩾ − κ n − 2 ∫ N − k ¯ R d V G \geqslant - {\kappa ^{n - 2}}\int _{N - } {{{\bar k}_R}\;dV} . These three estimates are sharp. For a ball in 3 3 -dimensional hyperbolic space, the ratio of the right-hand part of each estimate to its left-hand part (i.e. V ( κ 2 + 3 ) / 2 H V({\kappa ^2} + 3)/2H , V ( κ 2 + 1 ) / H V({\kappa ^2} + 1)/H and 2 κ V / G 2\kappa V/G respectively) approaches 1 as the radius → ∞ {\operatorname {radius}} \to \infty . The same ratios for the estimates \[ H ⩾ − 1 2 ( n − 1 ) ∫ N k s c d V and H ⩾ − 1 n − 1 ∫ N k ¯ R d V H \geqslant - \frac {1} {{2(n - 1)}}\int _N {{k_{sc}}\;} dV\quad {\text {and}}\quad H \geqslant - \frac {1} {{n - 1}}\int _N {{{\bar k}_R}\;dV} \] (rougher ones but without κ \kappa ) approach 3 4 \tfrac {3} {4} and 1 2 \tfrac {1} {2} respectively.