Deformations of CMC Surfaces Preserving the Hopf Differential
We show that any constant mean curvature (CMC) surface, either minimal or not, has a natural deformation through a whole family of CMC surfaces, all with the same Hopf differential, but different values of the mean curvature.
The family obtained depends on a choice of some fixed basepoint on the surface. However many surfaces, for example surfaces with symmetries with respect to some point, have a natural choice of basepoint, because these symmetries are preserved under the deformation.
Any minimal surface can be produced from holomorphic data, the socalled Weierstrass (or WeierstrassEnneper) representation. There is a generalization of the Weierstrass representation to nonminimal CMC surfaces, using loop groups. Given the Weierstrass data for a minimal surface, which is essentially a pair of holomorphic functions, and given a choice of basepoint, we give a formula in the reference below, for the generalized Weierstrass data for the associated nonminimal CMC surfaces. Images of the surfaces can then be computed using numerical methods.
More examples can be computed using this numerical implementation.
References:

Deformations of constant mean curvature surfaces preserving symmetries and the Hopf differential.
D. Brander and J. Dorfmeister. Preprint: arXiv:1302.2228 [math.DG]