Kähler-Einstein metrics and eigenvalue gaps (Q6628915)

Let \(M\) be a compact Kähler manifold with first Chern class \(c_1 (M) > 0\) and let \(\omega_0 \in c_1 (M)\) be a fixed metric. Mabuchi's \(K\)-energy [\textit{T. Mabuchi}, Tôhoku Math. J. (2) 38, No. 1--2, 575--593 (1986; Zbl 0619.53040)] is\N\[\NK_{\omega_0} (\omega ) = \frac{1}{V} \Big\{ \int_M \Big( \log \, \frac{\omega^n}{\omega_0^n} \Big) \, \omega^n +\N\]\N\[\N- \phi \sum_{j=0}^{n-1}\mathrm{Ric}({\omega_0} ) \, \omega^j \, \omega_0^{n-1-j} + \mu \, \sum_{j=0}^n \phi \, \omega^j \, \omega_0^{n-j} \Big\}\N\]\Nfor every metric \(\omega \in c_1 (M)\) of Kähler potential \(\phi \in C^\infty (M)\) with \(\omega_0 + \frac{i}{2} \, \partial \, \overline{\partial} \, \phi > 0\). If \(\overline{R} = \frac{1}{V} \int_M R \, \omega^n\) then the Euler-Lagrange equations of the variational principle \(\delta \, K_{\omega_0} (\omega ) = 0\) are \(R(\omega ) - \overline{R} = 0\) [\textit{D. H. Phong} and \textit{J. Sturm}, ``Lectures on stability and constant scalar curvature'', Preprint, \url{arXiv:0801.4179}, p. 11]. The Ricci potential \(u_\omega\) of a metric \(\omega \in C_1 (M)\) is determined by\N\[\N\mathrm{Ric} (\omega ) - \omega = - i \, \partial \, \overline{\partial} \, u_\omega \, , \;\;\; \int_M e^{- u_\omega} \, \omega^n = \int_M \omega^n \, .\N\]\NLet \(\lambda_\omega\) be the smallest strictly positive eigenvalue of \(\overline{\partial}\) on the space of complex vector fields of type \((1, 0)\) [\textit{D. H. Phong} et al., J. Differ. Geom. 81, No. 3, 631--647 (2009; Zbl 1162.32014)]. Let \(c_1 (M; \, A)\) be the set of all \(\omega \in c_1 (M)\) such that\N\[\N\| u_\omega \|_{C^0} + \| \nabla_\omega u_\omega \|_{C^0} + \| R_\omega \|_{C^0} \leq A \, , \;\;\; K_{\omega_0} (\omega ) \leq A \, .\N\]\NThe eigenvalue gap for \(c_1 (M ; \, A) \neq \emptyset\) is\N\[\N\lambda (M; \, A) = \inf_{\omega \in c_1 (M; \, A)} \lambda_\omega \, .\N\]\NAmong other results, the authors show that a compact Kähler manifold with \(c_1 (M) > 0\) and vanishing Futaki invariant admits a Kähler-Einstein metric if and only if \(\lambda (M; \, A) > 0\) for any \(A > 0\).