Tohoku Mathematical Journal

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

Peng Wang

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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 $Gr_{1,3}(\mathbb{R}^{1,n+3})=SO^+(1,n+3)/SO^+(1,3)\times SO(n)$ 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.

Article information

Source
Tohoku Math. J. (2), Volume 69, Number 1 (2017), 141-160.

Dates
First available in Project Euclid: 26 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1493172133

Digital Object Identifier
doi:10.2748/tmj/1493172133

Mathematical Reviews number (MathSciNet)
MR3640019

Zentralblatt MATH identifier
1368.53010

Subjects
Primary: 53A30: Conformal differential geometry
Secondary: 58E20: Harmonic maps [See also 53C43], etc. 53C43: Differential geometric aspects of harmonic maps [See also 58E20] 53C35: Symmetric spaces [See also 32M15, 57T15]

Keywords
Willmore surfaces normalized potential minimal surfaces Iwasawa decompositions

Citation

Wang, Peng. Willmore surfaces in spheres via loop groups III: on minimal surfaces in space forms. Tohoku Math. J. (2) 69 (2017), no. 1, 141--160. doi:10.2748/tmj/1493172133. https://projecteuclid.org/euclid.tmj/1493172133


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