On the dressing action of loop groups on constant mean curvature surfaces

We deal with the question of whether the Hopf differential on constant mean curvature surfaces parameterizes the dressing orbits of these surfaces. It is shown that in addition to the Hopf differential, there are infinitely many dressing invariants associated with an umbilic point of order larger than or equal to 2. Thus, when such an umbilic point is present, there are many dressing orbits sharing the same Hopf differential. We also give a procedure for computing all these dressing invariants associated with an umbilic point and they are used to show that the sizes of the dressing orbits in general depend on the topology on the loop group used.