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An extremal eigenvalue problem in Kähler geometry

We study Laplace eigenvalues $λ_k$ on Kähler manifolds as functionals on the space of Kähler metrics with cohomologous Kähler forms. We introduce a natural notion of a $λ_k$-extremal Kähler metric and obtain necessary and sufficient conditions for it. A particular attention is paid to the $λ_1$-extremal properties of Kähler-Einstein metrics of positive scalar curvature on manifolds with non-trivial holomorphic vector fields.