Infimum of the spectrum of Laplace-Beltrami operator on a bounded pseudoconvex domain with a Kähler metric of Bergman type (Q626506)

Let \((M,g)\) be a complete Kähler manifold of dimension \(n\). Let \(\lambda_1(\Delta_g,M)\) be the bottom of the spectrum of the Laplace-Beltrami operator. The authors consider the special case of a bounded pseudo-convex domain \(D\) in \(\mathbb C^n\) with \(C^2\)-smooth boundary. The authors find a class of Bergman-type metrics \(u\) on \(D\) such that \(\lambda_1(\Delta_u)=n^2\). Assume that the defining function \(r(z)\) belongs to \(C^2(\mathbb C^n)\) and that \(u(z)=-\log(-r(z))\) is strictly plurisubharmonic in \(D\); this function is used to define the Kähler potential giving the metric \[ g_u:=\sum_{i,j=1}^nu_{i\bar j}dz^i\otimes d\bar z^j\,. \] Set \[ |\partial u|_u^2=\sum_{i,j=1}^nu^{i\bar j}\partial_iu\partial_{\bar j}u \quad\text{and}\quad \beta(z)=\limsup_{w\rightarrow z}|\partial u(w)|^2_u \] and assume that \(\beta=1\) on \(\partial D\). (This is an analytic condition; the condition \(\beta=1\) near \(\partial M\) has a geometric interpretation related to the pseudo scalar curvature for a Kähler Einstein metric). Under these assumptions, the authors show: {\parindent6.5mm \begin{itemize}\item[(1)] \(\lambda_1(D)\leq\lambda_1(D-K)\leq n^2\) for any compact subset \(K\) of \(D\); \item[(2)] If \(r(z)\) is plursubharmonic in \(D\), then \(\lambda_1(D)=n^2\). \end{itemize}}