I was puzzeling with the pseudosphere , a surface that except for a cusp has a constant negative Gaussian curvature, but has not everywhere the same mean curvature. (https://en.wikipedia.org/wiki/Pseudosphere )
This made me wonder are there also surfaces that have (for a largish area) a constant positive Gaussian curvature but not a constant mean curvature?
Something like a surface that has principle curvatures $K_1=1 ,K_2=1$ for a single point or limited set of curves while for the other points in this neightboorhood the Gaussian curvature $K_1K_2 =1$ but $ K_1 \not = K_2$ and this not just at a small part of the surface but at a largish area, like an area bounded by a cusp or boundary.
The surface does not need to be closed.