Weierstrass-Kenmotsu representation of Willmore surfaces in spheres
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.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2019
- DOI:
- 10.48550/arXiv.1901.08395
- arXiv:
- arXiv:1901.08395
- Bibcode:
- 2019arXiv190108395D
- Keywords:
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- Mathematics - Differential Geometry
- E-Print:
- 21 pages. We split arXiv:1301.2756 according to referee's suggests. This is the first part. Comments are welcome