球面中不稳定的高阶极小Clifford超曲面低指标的刻画

A Characterization of Low Index of Unstable r-minimal Clifford Hypersurfaces in Spheres

根据单位球面中不稳定的高阶极小子流形的一个充分条件,构造了球面中一类不稳定的r-极小超曲面,即所谓的n维r-极小Clifford超曲面C1,n-1(r)=S1(r+1/n)(1/2)×Sn-1(n-r-1/n)(1/2),这里r是偶数,且r∈{0,1,…,n-1}.特别地,通过计算2-极小Clifford超曲面C1,n-1(2)的Jacobi算子的第二特征值,得到当n4时,其稳定性指标Ind2(C1,n-1(2))≥3n+3.

According to a sufficient condition of unstable higher-order minimal submanifolds in spheres, the authors con-struct a family of unstable r-minimal submanifolds, that is so-called r-minimal Clifford hypersurfaces Cl,n-1 （r）= S1（r＋1/n）（1/2）×Sn-1（n-r-1/n）（1/2）here r is evenand r∈{0,1,…,n-1}. In particular, by computing the second eigenvalue of Jacobi operatorof 2-minimal Clifford hypersurfaces C1,n-1 （2） , the authors obtain that when n≥ 4, the stability index Indz （C1,n-1（2））≥3n＋3.