Jacobi equations on minimal homogeneous submanifolds in homogeneous Riemannian spaces
For a minimal submanifold N in a Riemannian manifold M the second variation of the volume of N is given by an integral involving the Jacobi operator I acting on the sections in the normal bundle of N. The index of I is the sum of the multiplicities of the negative eigenvalues of I. The author treats the case where \(M=G/K\) is a Riemannian homogeneous space and \(N=H/K\cap H\). The Jacobi operator is calculated in algebraic terms. Now one can apply Frobenius reciprocity to decompose the space of sections in the normal bundle into finite dimensional I-invariant subspaces. These techniques are used to classify all compact subgroups of the classical compact Lie groups G that are stable minimal submanifolds. Here G carries the biinvariant metric. Finally, the space \(M=SU_{m+1}/T_ m\) is studied where \(T_ m\subset SU_{m+1}\) is a maximal torus. For certain N lower estimates for the index of I are derived.
scientific article; zbMATH DE number 4176040
scientific article