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arXiv · 1901.08395

Weierstrass-Kenmotsu representation of Willmore surfaces in spheres

Josef F. Dorfmeister·Peng Wang
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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 to a Willmore surface. The exceptional case will also be discussed.

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Josef F. Dorfmeister, Peng Wang. 2019-01-24. Weierstrass-Kenmotsu representation of Willmore surfaces in spheres. https://arxiv.org/abs/1901.08395

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