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We study Laplace eigenvalues $\lambda_k$ on K\"ahler manifolds as functionalson the space of K\"ahler metrics with cohomologous K\"ahler forms. We introducea natural notion of a $\lambda_k$-extremal K\"ahler metric and obtain necessaryand sufficient conditions for it. A particular attention is paid to the$\lambda_1$-extremal properties of K\"ahler-Einstein metrics of positive scalarcurvature on manifolds with non-trivial holomorphic vector fields.

An extremal eigenvalue problem in K\"ahler geometry