On conditions for existence of umbilical points on a convex surface (Q1921830)

Let \(F\) be a connected complete \(C^\infty\) surface homeomorphic to a plane in the Euclidean space \(\mathbb{R}^3\) and \(k_1\), \(k_2\) its principal curvatures. The author establishes the following result. If the surface \(F\) satisfies one of the following conditions: (i) the integral curvature is strictly smaller than \(2\pi\), (ii) the Gauss curvature and the modules of the gradient functions for \(k_1\) and \(k_2\) are bounded on \(F\); then \(\inf_{p\in F} (k_2(p)-k_1(p))=0\).