The umbilic set of Willmore surfaces

It is well known that the umbilic points of minimal surfaces in spaces of constant sectional curvature consist only of isolated points unless the surface is totally umbilic on some connected component, as for example the Hopf form is holomorphic. In this note, we prove that on Willmore surfaces in codimension one the umbilic set is locally a one dimensional real-analytic manifold without boundary or an isolated point.