An extremal eigenvalue problem in Kähler geometry

We study Laplace eigenvalues λkλk on Kähler manifolds as functionals on the space of Kähler metrics with cohomologous Kähler forms. We introduce a natural notion of a λkλk-extremal Kähler metric and obtain necessary and sufficient conditions for it. A particular attention is paid to the λ1λ1-extremal properties of Kähler–Einstein metrics of positive scalar curvature on manifolds with non-trivial holomorphic vector fields.