Deformation of K\"{a}hler Metrics and an Eigenvalue Problem for the Laplacian on a Compact K\"{a}hler Manifold

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