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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Inequalities of Gauss-Bonnet type for a convex domain


Author: B. V. Dekster
Journal: Proc. Amer. Math. Soc. 86 (1982), 632-637
MSC: Primary 53C20; Secondary 53C40, 53C45
MathSciNet review: 674095
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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

$\displaystyle H \geqslant \frac{{n - 2}} {2}{\kappa ^2}V - \frac{1} {{2(n - 1)}... ...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

$\displaystyle 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.

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DOI: https://doi.org/10.1090/S0002-9939-1982-0674095-3
Article copyright: © Copyright 1982 American Mathematical Society