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.

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Weierstrass-Kenmotsu 表示球体中的威尔莫尔曲面

威尔莫尔曲面$y:M\rightarrow S^{n+2}$有一个自然谐波取向的共形高斯图$Gr_{y}:M\rightarrow SO^{+}(1,n+3)/SO(1,3)\times SO(n)$，它映射每个点$p\in M$到其定向平均曲率 2 球体$p$. 一个简单的观察表明，Willmore 曲面的所有共形高斯图都满足一个受限的幂等性条件，这将被称为“强共形谐波”。本文的目的是从黎曼曲面表征那些强共形谐波映射$M$到$SO^{+}(1,n+3)/SO^{+}(1,3)\times SO(n)$, 这是一些 Willmore 曲面的保形高斯图$S^{n+2}.$事实证明，一般来说，强共形调和的条件足以与 Willmore 曲面相关联。特殊情况也将讨论。