Deformation of Kähler metrics and an eigenvalue problem for the Laplacian on a compact Kähler manifold

We study an eigenvalue problem for the Laplacian on a compact Kähler manifold. Considering the k-th eigenvalue \(\lambda _{k}\) ${\lambda }_k$ as a functional on the space of Kähler metrics with fixed volume on a compact complex manifold, we introduce the notion of \(\lambda _{k}\) ${\lambda }_k$ -extremal Kähler metric. We deduce a condition for a Kähler metric to be \(\lambda _{k}\) ${\lambda }_k$ -extremal. As examples, we consider product Kähler manifolds, compact isotropy irreducible homogeneous Kähler manifolds and flat complex tori.