Willmore surfaces in spheres via loop groups III: on minimal surfaces in space forms

Abstract

The family of Willmore immersions from a Riemann surface into $S^{n+2}$ can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in ${\mathbb{R}}^{n+2}$ and those which are not conformally equivalent to a minimal surface in ${\mathbb{R}}^{n+2}$. On the level of their conformal Gauss maps into $G{r}_{1,3}({\mathbb{R}}^{1,n+3})=S{O}^{+}(1,n+3)/S{O}^{+}(1,3)\times SO(n)$ these two classes of Willmore immersions into $S^{n+2}$ correspond to conformally harmonic maps for which every image point, considered as a 4-dimensional Lorentzian subspace of ${\mathbb{R}}^{1,n+3}$, contains a fixed lightlike vector or where it does not contain such a "constant lightlike vector". Using the loop group formalism for the construction of Willmore immersions we characterize in this paper precisely those normalized potentials which correspond to conformally harmonic maps containing a lightlike vector. Since the special form of these potentials can easily be avoided, we also precisely characterize those potentials which produce Willmore immersions into $S^{n+2}$ which are not conformal to a minimal surface in ${\mathbb{R}}^{n+2}$. It turns out that our proof also works analogously for minimal immersions into the other space forms.

20 pages. Revised Version

References in corpus (3)

• Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms
• Harmonic maps of finite uniton type into inner symmetric spaces
• Willmore surfaces in spheres via loop groups II: a coarse classification of Willmore two-spheres by potentials

Cited by in corpus (2)

• Willmore surfaces in spheres via loop groups II: a coarse classification of Willmore two-spheres by potentials
• Willmore surfaces in spheres: the DPW approach via the conformal Gauss map