Let us consider the tractroid (pseudosphere) obtained by rotation from the tractrix curve. The surface is not defined on the "big rim", so it is not a complete set. Hilbert's theorem states that there exists no complete regular surface of constant negative Gaussian curvature immersed in $\mathbb{R}^{3}$.
Then, if we truncate the tractroid surface with two planes that are orthogonal to the rotation axis, apparently we obtain a compact surface with negative curvature... but that it not possible by Hilbert's theorem. Is there a loss of regularity along the borders?