WEIERSTRASS–KENMOTSU REPRESENTATION OF WILLMORE SURFACES IN SPHERES
WEIERSTRASS–KENMOTSU REPRESENTATION OF WILLMORE SURFACES IN SPHERES
复制标题
DOI：
10.1017/nmj.2020.6
复制
发表时间：
2019-01
影响因子：
0.8
通讯作者：
J. Dorfmeister;Pengfu Wang
中科院分区：
文献类型：
--
作者：
J. Dorfmeister;Pengfu Wang
Willmore 曲面 $y:M\rightarrow S^{n+2}$ 具有自然谐波定向共形高斯映射 $Gr_{y}:M\rightarrow SO^{+}(1,n+3)/SO(1,3)\times SO(n)$，它将每个点 $p\in M$ 映射到其在 $p$ 处的定向平均曲率 2 球面。一个简单的观察表明，威尔莫尔曲面的所有共形高斯图都满足受限幂零条件，这将被称为“强共形谐波”。本文的目标是表征从黎曼曲面 $M$ 到 $SO^{+}(1,n+3)/SO^{+}(1,3)\times SO(n)$ 的强共形调和映射，它们是 $S^{n+2}$ 中某些 Willmore 曲面的共形高斯映射。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。还将讨论特殊情况。
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.