On boundedness of Gaussian curvature from zero on simplest hyperbolic surfaces (Q677758)

Let \(F\) be a simple hyperbolic surface in \(E^3\) (in the sense of \textit{A. L. Verner} [Mat. Sb. 75(117), 112-139 (1968; Zbl 0164.21602)]), given by \(z=z(x,y)\), \(z^2_x+ z^2_y< \infty\). Its normal representation \(F^*\) on the \((p,q)\)-plane is given by \(p= z_x(x, y)\), \(q= z_y(x,y)\) (see \textit{I. Ja. Bakel'man}, \textit{A. L. Verner} and \textit{B. E. Kantor} [Introduction to differential geometry ``in the large'' (Nauka, Moscow) (1973; Zbl 0276.53039)]). It is proved that for each regular point \(A^*_\infty \in\partial F^*\) there is no neighborhood \(V_{A^*_\infty}\) of \(A^*_\infty\) such that the Gaussian curvature is bounded away from zero on the inverse image of \(V_{A^*_\infty}\) in \(E^3\).