Compact surfaces with boundary of constant negative curvature

Question

Consider a surface (with boundary) diffeomorphic to $S^1 \times [0, 1]$ and with constant negative curvature, sitting inside $\mathbb{R}^3$. All the examples I know of such surfaces are "part of" (or "cut out of", if that makes better sense) a surface of revolution (examples are the catenoid, tractricoid, etc.; O'Neill's "Elementary Differential Geometry" has a comprehensive list on page 261, Exercise 7). I was wondering, are all constantly negatively curved $S^1 \times [0, 1]$ obtained this way (that is, cut out from a surface of revolution)?

Answer 1

Apart from three types of negative Gauss curvature which are surfaces of revolution ( Beltrami central, hyper,hypo varieties ) we have the Breather, Kuen, Kink surface, Dini are not part of surfaces of revolution. The last two are twisted from the axisymmetric surfaces by modifying the parametrization or ode.