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

TL;DR

This paper systematically studies symmetric Willmore surfaces in spheres with orientation reversing symmetries, providing new examples and conditions for their projection to real projective planes, expanding understanding of these geometric objects.

Contribution

It introduces a comprehensive framework for analyzing orientation reversing symmetric Willmore surfaces and presents new examples, including previously unknown immersions from real projective planes.

Findings

Derived a necessary condition for isotropic Willmore surfaces to descend to $\,\mathbb{R}P^2$

Constructed new examples of Willmore immersions from $\,\mathbb{R}P^2$ to $S^4$

Extended the theory to include orientation reversing symmetries in Willmore surfaces.

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. 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 $\mathbb{R} P^2$ to $S^4$. The Veronese sphere and several other examples of non-branched Willmore immersions from $\mathbb{R} P^2$ to $S^4$ are derived as an illustration of the general theory. The Willmore immersions of $\mathbb{R} P^2$, just mentioned and different from the Veronese sphere, are new to the authors' best knowledge.