An extremal eigenvalue problem in K\"ahler geometry

We study Laplace eigenvalues ${\lambda }_k$ on K"ahler manifolds as functionals on the space of K"ahler metrics with cohomologous K"ahler forms. We introduce a natural notion of a ${\lambda }_k$-extremal K"ahler metric and obtain necessary and sufficient conditions for it. A particular attention is paid to the ${\lambda }_1$-extremal properties of K"ahler-Einstein metrics of positive scalar curvature on manifolds with non-trivial holomorphic vector fields.