Inequalities of Gauss-Bonnet type for a convex domain
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- by B. V. Dekster PDF
- Proc. Amer. Math. Soc. 86 (1982), 632-637 Request permission
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.References
- B. V. Dekster, An inequality of isoperimetric type for a domain in a Riemannian space, Mat. Sb. (N.S.) 90(132) (1973), 257–274, 326 (Russian). MR 0362159
- B. V. Dekster, Estimates of the length of a curve, J. Differential Geometry 12 (1977), no. 1, 101–117. MR 470906
- B. V. Dekster, The volume of a slightly curved submanifold in a convex region, Proc. Amer. Math. Soc. 68 (1978), no. 2, 203–208. MR 474147, DOI 10.1090/S0002-9939-1978-0474147-6
- B. V. Dekster, Upper estimates of the length of a curve in a Riemannian manifold with boundary, J. Differential Geometry 14 (1979), no. 2, 149–166. MR 587544
- D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Grossen, Lecture Notes in Mathematics, No. 55, Springer-Verlag, Berlin-New York, 1968 (German). MR 0229177
- Hanno Rund, Invariant theory of variational problems on subspaces of a Riemannian manifold, Hamburger Mathematische Einzelschriften (N.F.), No. 5, Vandenhoeck & Ruprecht, Göttingen, 1971. MR 0343149
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 86 (1982), 632-637
- MSC: Primary 53C20; Secondary 53C40, 53C45
- DOI: https://doi.org/10.1090/S0002-9939-1982-0674095-3
- MathSciNet review: 674095