Some properties of surfaces with slowly varying negative extrinsic curvature in a Riemannian space

We consider surfaces of negative extrinsic curvature in a Riemannian space with nonpositive curvature. We prove that the following inequality holds on a surface which is complete in the sense of the intrinsic metric:
$$ \sup_F\biggl\{\biggl|\operatorname{grad}\frac1k\biggr|+\frac{\Lambda-\lambda}{2k^2}\biggr\}=q>\frac1{\sqrt3}, $$
here $F$ is the surface being considered, $k=\sqrt{K_e}$ ($K_e$ is the extrinsic curvature of $F$) and $\Lambda$ and $\lambda$ are the maximum and minimum of the Riemannian curvature of the space $R$ at a given point.
This theorem generalizes a theorem of Efimov concerning $q$-metrics. We give an example of a surface for which $q=4,5$.
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