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2017 Willmore surfaces in spheres via loop groups III: on minimal surfaces in space forms
Peng Wang
Tohoku Math. J. (2) 69(1): 141-160 (2017). DOI: 10.2748/tmj/1493172133

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.

Citation

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Peng Wang. "Willmore surfaces in spheres via loop groups III: on minimal surfaces in space forms." Tohoku Math. J. (2) 69 (1) 141 - 160, 2017. https://doi.org/10.2748/tmj/1493172133

Information

Published: 2017
First available in Project Euclid: 26 April 2017

zbMATH: 1368.53010
MathSciNet: MR3640019
Digital Object Identifier: 10.2748/tmj/1493172133

Subjects:
Primary: 53A30
Secondary: 53C35, 53C43, 58E20

Rights: Copyright © 2017 Tohoku University

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Vol.69 • No. 1 • 2017
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