Deformation of Kähler metrics and an eigenvalue problem for the Laplacian on a compact Kähler manifold (Q6636596)

Let \((M,J,g,\omega)\) be a compact Kähler manifold. In this paper, \(g\) is called \(\lambda_k\)-extremal if \N\[\N\left( \left.\frac{d}{dt}\right|_{t=0^-}\lambda_k(g_t) \right) \left( \left.\frac{d}{dt}\right|_{t=0^+}\lambda_k(g_t) \right) \leq 0 \N\]\Nholds for any \(1\)-parameter family of volume-preserving Kähler metrics \((g_t)_t\) compatible with \((M,J)\) that depends real analytically on \(t\), where \(\lambda_k(g_t)\) denotes the \(k\)-eigenvalue of the Laplace-Beltrami operator of \((M,g_t)\).\N\NThe main theorem ensures that \(g\) is \(\lambda_1\)-extremal if and only if there exists a finite collection of eigenfunctions \(\{f_j\}_{j=1}^N\) such that \N\[\N H\left(\sum_{j=1}^N f_jd d^c f_j\right)=-\omega \N\]\Nand \N\[\N\lambda_1(g)^2\left(\sum_{j=1}^N f_j^2\right) - 2\lambda_1(g) \left(\sum_{j=1}^N |\nabla f_j|^2 \right) +\sum_{j=1}^N |dd^cf_j|^2 =0.\N\]\NHere, \(H\) is the harmonic projector, which is defined using the Hodge decomposition on the compact Kähler manifold.\N\NIn addition, the author proves that the metric on a compact isotropy irreducible Kähler manifold is \(\lambda_1\)-extremal, and constructs an example of a Kähler metric that is \(\lambda_1\)-extremal within its Kähler class, but not so for all volume-preserving deformations of the Kähler metric. The final section is dedicated to flat complex tori.\N\NA weaker notion of \(\lambda_k\)-extremality was defined and studied in [\textit{V. Apostolov} et al., J. Geom. Phys. 91, 108--116 (2015; Zbl 1325.49054)]. The introductory section of the paper under review makes a good comparison between their results and many others.