Let G be a compact Lie group
of isometries acting on a riemannian manifold M. In recent years, there has been a
great deal of interest in minimal submanifolds that arise as orbits of such an action.
In this paper, we formulate necessary and sufficient conditions for the stability of
minimal codimension two principal orbits. These conditions are expressed
in terms of the eigenvalues of a G-invariant vector field on the orbit, the
eigenvalues of the laplacian of the orbit, and the eigenvalues of the hessian of
the volume function. Next we use a poincaré inequality along with the
orthogonality relations on the group G to find conditions for the stability of
exceptional orbits. These conditions are used to find new examples of stable
minimal submanifolds in the generalized lens-spaces and the quaternionic space
forms.
