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