S^4中具有调和共形高斯映照的超曲面

Hypersurfaces with Harmonic Conformal Gauss Map in S^4

设x:M^n→S^(n+1)是球面S^(n+1)中的一个定向超曲面,其共形高斯映照G=(H,Hx+en+.1):M^n→R_1S^(n+3)是M(o|¨)bius变换群下的一个不变量,其中H,e(n+1)+1分别是超曲面x的平均曲率和单位法向量场.本文研究了S^4中具有调和共形高斯映照的超曲面,分类了具有调和共形高斯映照和常M(o|¨)bius数量曲率的超曲面,给出了具有调和共形高斯映照但不是Willmore超曲面的例子.

Let x:M^n → S^(n+1) be an oriented hypersurface in S^(n+1),the conformal Gauss map G =(H,H_x.+ e_(n+1)):M^n→ R_1^(n+3) is invariant under Mobius transformations of S^(n+1),where H,e_(n+1) are the mean curvature,the global unit normal vector field of x,respectively.In this paper,we study the oriented hypersurface x:M^3 → S^4with harmonic conformal Gauss map,and we classify the hypersurfaces in S^4 with constant Mobius scalar curvature under Mobius transformation group,which gives some examples of hypersurfaces with harmonic conformal Gauss map,but not Willmore hypersurfaces.