On symmetric Willmore surfaces in spheres II: The orientation reversing case

Abstract

In this paper we provide a systematic treatment of Willmore surfaces with orientation reversing symmetries and illustrate the theory by (old and new) examples. For finite order orientation reversing symmetries with fixed point as well as for finite order orientation reversing symmetries without fixed point the general theory is presented and concrete examples are given.Moreover, we apply our theory to isotropic Willmore two-spheres in S 4 and derive a necessary condition for such ( possibly branched) isotropic surfaces to descend to (possibly branched) maps from RP 2 to S 4 . The Veronese sphere and several other examples of non-branched Willmore immersions from RP 2 to S 4 are derived as an illustration of the general theory. The Willmore immersions of RP 2 , just mentioned and different from the Veronese sphere, are new to the authors' best knowledge. All these examples are conformally equivalent to minimal immersions into S 4 . We also provide a characterization of equivariant Willmore Moebius strips in S n+2 , and relate our results to Lawson's minimal Moebius strip in S 3 . Moreover, we present a defining characterization of all equivariant Willmore Klein bottles. Finally, we discuss orientation reversing symmetries of finite order which have a fixed point in the surface.

Willmore deformations between minimal surfaces in $H^{n+2}$ and $S^{n+2}$

“…4 . In this subsection, we consider some non-oriented minimal surfaces in H 4 , which is based on the work of [20] and [54]. Set…”