On the Kähler manifolds with the largest infimum of spectrum of Laplace-Beltrami operators and sharp lower bound of Ricci or holomorphic bisectional curvatures (Q641517)

Let \(\mathcal D(A)\) be the ellipsoid \(\mathcal D(A)=\{z\in\mathbb C^n, |z|^2+\text{Re}\sum_{j=1}^nA_jz_j^2<1\}\) with \(0\leq A_1\leq A_2\leq\ldots\leq A_n< 1\). The author analyses two types of complete Kähler metrics \(g_u\) with potential \(u\) on \(\mathcal D(A)\): the unique Kähler-Einstein metric [\textit{S.-Y. Cheng} and \textit{S.-T. Yau}, Commun. Pure Appl. Math. 33, 507--544 (1980; Zbl 0506.53031)] with \(\text{Ric}_{g_u}=-2(n+1)\) and another one with holomorphic bisectional curvature at least \(-1\) whose existence is established in case of the pinching \(A_n\leq 2/5\). For both metrics, the author shows that the boundary defining function \(\rho=-e^{-u}\) associated to the Kähler potential \(u\) is plurisubharmonic on the bounded convex domain \(\mathcal D(A)\). Let \((M,g)\) an \(n\)-dimensional complete Riemannian manifold and \[ \lambda(M,g)=\inf_{\varphi\in\mathcal C^\infty_0(M)}[\int_M\nabla\varphi\|^2dv_g/\int_M\|\varphi\|^2dv_g] \] be the infimum of the spectrum of the Laplace-Beltrami operator. The upper bound \(\lambda(M,g)\leq n^2\) has been proved either for metrics with bisectional curvature at least \(-1\) by \textit{P. Li} and \textit{J. Wang} [J. Differ. Geom. 69, No. 1, 43--74 (2005; Zbl 1087.53067)] or for Kähler-Einstein metrics with \(\text{Ric}_g=-2(n+1)\) by \textit{O. Munteanu} [J. Differ. Geom. 83, No. 1, 163--187 (2009; Zbl 1183.53068)]. Moreover, if the manifold \(M\) is a pseudo-convex domain of the complex space \(\mathbb C^n\), \textit{M.-A. Tran} and the author [Commun. Anal. Geom. 18, No. 2, 375--395 (2010; Zbl 1215.58016)] proved that \((M,g_u)\) is maximal with \(\lambda(M,g_u)=n^2\) if the negative defining function \(\rho=-e^{-u}\) is plurisubharmonic. Hence, this metrics family on \(\mathcal D(A)\) shows that the maximisation of the spectrum bottom \(\lambda(M,g)\) does not induce necessarily rigidity for its maximizing domains (e.g. the complex hyperbolic space realized as the complex ball \(\mathcal D(0)\)). In fact, two different ellipsoids of the family \(\mathcal D(A)\) are never biholomorphic equivalent, a fact proved by \textit{S. M. Webster} [Invent. Math. 43, 53--68 (1977; Zbl 0348.32005)].