|
This article is cited in 3 scientific papers (total in 3 papers)
A geometric property of extremal surfaces
É. I. Kovalevskaya Institute of Mathematics, Academy of Sciences Byelorussian SSR
Abstract:
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.
Received: 17.12.1975
Citation:
É. I. Kovalevskaya, “A geometric property of extremal surfaces”, Mat. Zametki, 23:2 (1978), 177–181; Math. Notes, 23:2 (1978), 99–101
Linking options:
https://www.mathnet.ru/eng/mzm8131 https://www.mathnet.ru/eng/mzm/v23/i2/p177
|
Statistics & downloads: |
Abstract page: | 169 | Full-text PDF : | 68 | First page: | 1 |
|