Inequalities of Gauss-Bonnet type for a convex domain

Abstract:

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