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WEIERSTRASS–KENMOTSU REPRESENTATION OF WILLMORE SURFACES IN SPHERES

Published online by Cambridge University Press:  25 February 2021

JOSEF F. DORFMEISTER
Affiliation:
Fakultät für Mathematik, Technische Universität München, Boltzmannstr.3, D-85747, Garching, Germany email dorfm@ma.tum.de
PENG WANG
Affiliation:
College of Mathematics and Informatics, FJKLMAA, Fujian Normal University, Qishan Campus, Fuzhou 350117, PR China email pengwang@fjnu.edu.cn
Corresponding

Abstract

A Willmore surface $y:M\rightarrow S^{n+2}$ has a natural harmonic oriented conformal Gauss map $Gr_{y}:M\rightarrow SO^{+}(1,n+3)/SO(1,3)\times SO(n)$ , which maps each point $p\in M$ to its oriented mean curvature 2-sphere at $p$ . An easy observation shows that all conformal Gauss maps of Willmore surfaces satisfy a restricted nilpotency condition, which will be called “strongly conformally harmonic.” The goal of this paper is to characterize those strongly conformally harmonic maps from a Riemann surface $M$ to $SO^{+}(1,n+3)/SO^{+}(1,3)\times SO(n)$ , which are the conformal Gauss maps of some Willmore surface in $S^{n+2}.$ It turns out that generically, the condition of being strongly conformally harmonic suffices to be associated with a Willmore surface. The exceptional case will also be discussed.

Type
Article
Copyright
© 2020 Foundation Nagoya Mathematical Journal

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Footnotes

The second author is thankful to the ERASMUS MUNDUS TANDEM Project for the financial supports to visit the TU München.

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WEIERSTRASS–KENMOTSU REPRESENTATION OF WILLMORE SURFACES IN SPHERES
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