Comparison surfaces for the Willmore problem

The infimum of the conformally invariant functional W = / H 2 is estimated for each regular homotopy class of immersed surfaces in R 3 . Consequently, we obtain rather sharp bounds on the maximum multiplicity and branching order of a f -minimizing surface. In the case of RP 2 we provide an example of a symmetric ^-minimizing Boy's surface (W -\2)-as well as symmetric static surfaces of higher index-thereby solving part of the Willmore problem.