Sasaki manifolds, Kähler cone manifolds and biharmonic submanifolds

Abstract

For a Legendrian submanifold $M$ of a Sasaki manifold $N$, we study harmonicity and biharmonicity of the corresponding Lagrangian cone submanifold $C(M)$ of a Kähler manifold $C(N)$. We show that, if $C(M)$ is biharmonic in $C(N)$, then it is harmonic; and $M$ is proper biharmonic in $N$ if and only if $C(M)$ has a nonzero eigen-section of the Jacobi operator with the eigenvalue $m=\dim M$.