Conformal minimal two-spheres in Q_n  

作　　者：JIAO XiaoXiang WANG Jun 

机构地区：[1]School of Mathematics,Graduate University,Chinese Academy of Sciences,Beijing 100049,China

出　　处：《Science China Mathematics》2011年第4期817-830,共14页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No.11071248);Knowledge Innovation Funds of CAS (Grant No.KJCX3-SYW-S03);the President Fund of GUCAS

摘　　要：In this paper we study,using moving frames,conformal minimal two-spheres S2 immersed into a complex hyperquadric Qn equipped with the induced Fubini-Study metric from a complex projective n+1-space CPn+1.Two associated functions τX and τY are introduced to classify the problem into several cases.It is proved that τX or τY must be identically zero if f:S2 → Qn is a conformal minimal immersion.Both the Gaussian curvature K and the Khler angle θ are constant if the conformal immersion is totally geodesic.It is also shown that the conformal minimal immersion is totally geodesic holomorphic or antiholomorphic if K = 4.Excluding the case K = 4,conformal minimal immersion f:S2 → Q2 with Gaussian curvature K2 must be totally geodesic with(K,θ) ∈ {(2,0),(2,π/2),(2,π)}.In this paper we study,using moving frames,conformal minimal two-spheres S2 immersed into a complex hyperquadric Qn equipped with the induced Fubini-Study metric from a complex projective n＋1-space CPn＋1.Two associated functions τX and τY are introduced to classify the problem into several cases.It is proved that τX or τY must be identically zero if f：S2 → Qn is a conformal minimal immersion.Both the Gaussian curvature K and the Khler angle θ are constant if the conformal immersion is totally geodesic.It is also shown that the conformal minimal immersion is totally geodesic holomorphic or antiholomorphic if K = 4.Excluding the case K = 4,conformal minimal immersion f：S2 → Q2 with Gaussian curvature K2 must be totally geodesic with（K,θ） ∈ {（2,0）,（2,π/2）,（2,π）}.

关 键 词：complex hyperquadric Gaussian curvature Khler angle minimal immersion totally geodesic 

分 类 号：O186.11[理学—数学] TP273[理学—基础数学]