Bottom of the Spectrum of Complete K\"ahler Metrics from Finite-Mass Plurisubharmonic Exhaustions

Abstract

Let $\Omega\subset\mathbb{C}^{n}$ be a bounded domain, and let $\rho:\Omega\to[-1,0)$ be a smooth strictly plurisubharmonic exhaustion function. We consider the logarithmic potential $g=-\log(-\rho)$ and the associated complete K\"ahler metric $\omega=dd^{c}g$. We prove that if $\rho$ satisfies the finite weighted Monge--Amp\`ere mass condition $\int_{\Omega}(-\rho)^{\varepsilon}(dd^{c}\rho)^{n}<+\infty$ for every $\varepsilon>0$, then the bottom of the spectrum of the Laplace--Beltrami operator of $(\Omega,\omega)$ satisfies $\lambda_{0}(\Delta_{\omega},\Omega)=n^{2}$. The lower bound follows from the standard estimate applied to $g$, together with the inequality $|\partial g|_{\omega}^{2}\le 1$. For the reverse inequality, for each $\alpha>n/2$, we set $f=(-\rho)^{\alpha}$ and prove that $f\in W^{1,2}(\Omega,\omega)$ if and only if $\int_{\Omega}(-\rho)^{2\alpha-n}(dd^{c}\rho)^{n}<+\infty$. Under the finite weighted Monge--Amp\`ere mass condition, this allows us to let $\alpha\downarrow n/2$ in the Rayleigh quotient and obtain the upper bound $\lambda_{0}(\Delta_{\omega},\Omega)\le n^{2}$. As an application, Cegrell's theorem gives a smooth strictly plurisubharmonic exhaustion with finite Monge--Amp\`ere mass on every bounded hyperconvex domain; the associated complete K\"ahler metric constructed from this exhaustion therefore satisfies $\lambda_{0}(\Delta_{\omega},\Omega)=n^{2}$.