球面上极小子流形和全脐子流形的新特征

NEW CHARACTERIZATIONS OF MINIMAL SUBMANIFOLDS AND TOTALLY UMBILICAL SUBMANIFOLDS IN SPHERES

定义两个Ｓｃｈｒｏｄｉｎｇｅｒ算子Ｌ１、Ｌ２，先详细研究球面Ｓｎ＋ｐ中的极小子流形和全脐子流形，然后由算子Ｌ１和Ｌ２的第一特征值的估计给出Ｃｌｉｆｏｒｄ环。

Two Schrdinger operators are first defined. We carefully study the submanifolds in spheres, then a new characterization of Clifford tori in S n+1 and Veronese surface in S 4 is given by the first eigenvalue of the Schrdinger operator L 1=-Δ+ap+(1-p)pS . When the mean curvature vector of submanifold M is parallel but M is not geodesic in S n+p , we also characterise small n spheres of n+p dimensional Euclid spheres by the first eigenvalue of the Schrdinger operator L 2=-Δ-(1-4n)σ .