Geometry of Sasaki manifolds, Kähler cone manifolds and bi-harmonic submanifolds

For a Legendrian submanifold M of a Sasaki manifold N, we study harmonicity and bi-harmonicity of the corresponding Lagrangian cone submanifold C(M) of a Kähler manifold C(N). We show that, if C(M) is bi-harmonic in C(N), then it is harmonic; and M is proper bi-harmonic in N if and only if C(M) has a non-zero eigen-section of the Jacobi operator with the eigenvalue m=dim⁡M. For more details, see [34].