A~geometric property of extremal surfaces

É. I. Kovalevskaya

Matematičeskie zametki, Tome 23 (1978) no. 2, pp. 177-181.

Résumé

Let the surface $\Gamma\in R^3$ be defined by the equation $z=f(x,y)$, where $f(x,y)$ is a function 3 times continuously differentiable in $R^2$. It is proved that if the total (Gaussian) curvature of the surface $\Gamma$ is nonzero almost everywhere on $\Gamma$ (in the sense of Lebesgue measure in $R^2$), then $\Gamma$ is extremal, i.e., for almost all $(x,y)\in R^2$ the inequality $$ \max(\|qx\|,\|qy\|,\|qf(x,y)\|)>q^{-1/3-\varepsilon}, $$ holds for all integral $q\ge q_0(f)$, where $\|x\|$ is the distance from the real number $x$ to the nearest integer and $\varepsilon>0$ is arbitrarily small.