Comparison surfaces for the Willmore problem (Q1101980)

The infimum of the conformally invariant functional \(W=\int H\) 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 W-minimizing surface. In the case of RP 2 we provide an example of a symmetric W-minimizing Boy's surface \((W=12\pi)\), as well as symmetric static surfaces of higher index, thereby solving part of the Willmore problem.