Can we construct a non-trivial bicubic Bezier surface with mean curvature = 0 everywhere?

Question

If possible how to display the min and max mean curvature values?

Answer 1

Yes. The Enneper surface is a minimal surface, so its mean curvature is everywhere zero, and it has a bicubic parameterization. If you take the parametric equations given on that Wikipedia page, and restrict the parameter values to $(u,v) \in [0,1] \times [0,1]$, then you get a bicubic patch that is a rectangular region of the surface. Expressing this patch in Bezier form is just a change of basis.

In fact Cosin and Monterde showed that all minimal bicubic Bezier surfaces are affine transforms of a portion of the Enneper surface:

C. Cosın, and J. Monterde, Bezier surfaces of minimal area, Proc. Int. Conf. of Comput. Sci. ICCS 2002, LNCS 2330, Springer, Berlin Heidelberg, 2002, 72–81

To find out how to calculate and display mean curvature in Mathematica, look at Gray's book "Modern Differential Geometry of Curves and Surfaces".