Variational characterizations of invariant submanifolds in Sasaki manifolds

Abstract

In this note, we generalize our previous results in [4] and [5] from Kähler manifold to Sasaki manifold. We prove that for a submanifold tangent to the Reeb vector field in a Sasaki manifold, the area functional, as a functional of the ambient metrics, is stationary if and only if it is an invariant submanifold. We also show that if the submanifold has dimension 3, then it is basic stable if and only if it is an invariant submanifold.