Hilbert theorem and constant negative curvature surfaces 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?

It's compact, but obviously not _complete_.

(Wait, are you including the truncating surfaces in the result? Then it fails to be differentiable at the edges, so it doesn't have a Gaussian curvature there, much less a constant negative one).