Estimates of the curvature of a~three-dimensional evolute

Yu. A. Volkov ; B. V. Dekster

Sbornik. Mathematics, Tome 12 (1970) no. 4, pp. 615-637

Résumé

The article discusses compact three-dimensional evolutes of positive curvature and convex boundary and establishes inequalities that connect their integral characteristics: volume $V$, boundary area $S$, mean integral curvature of the boundary $H$, radius of the inscribed sphere $r$, and inner integral curvature $\Omega$. The last characteristic is a measure of non-Euclidicity of an evolute involved: $\Omega=0$ if and only if the evolute is locally Euclidean. The inequalities obtained in particular imply that $2\pi\chi r\leqslant H+\Omega$, where $\chi$ is the Euler characteristic of the evolute boundary. For an evolute homeomorphic to a sphere we have $\chi=2$, so that $r\leqslant\frac{H+\Omega}{4\pi}$, $V\leqslant Sr\leqslant\frac{H+\Omega}{4\pi}$. Equality in the estimate $r\leqslant\frac{H+\Omega}{4\pi}$ is achieved for a Euclidean sphere: for it $\Omega=0$ and $r=\frac H{4\pi}$. Figures: 6. Bibliography: 2 titles.