Bianchi Surfaces
Surfaces are called Weingarten surfaces (or just Weingarten for short), if for the Gaussian curvature K and the mean curvature function H
d K ∧ d H = 0
holds on all the surface. In what follows, the principle curvatures are functional related, i.e locally one of the principle curvature can be derived as a function of the other or vice versa. The animation below shows a deformation that acts on the set of Bianchi surfaces which are Weingarten surfaces.
Such surfaces can be described in terms of the solutions of a Painleve V equation and the deformation parameter controlling the shown deformation is a parameter in this equation.
In particular all Bianchi surfaces of revolution with non-constant curvature are either the catenoid ( or homotheisis of it) or can be described in terms of Painleve V functions.
The corresponding associated family of each surface you see in the animation is given by a scaling of only one surfaces, which you do not see.
Here is an example of such a family:
© Ulrich Eitner   -   email to: ulisfb288.math.tu-berlin.de