Bottom of the Spectrum of Complete Kähler Metrics from Finite-Mass Plurisubharmonic Exhaustions

Let $\mathrm{\Omega }\subset {\mathbb{C}}^n$ be a bounded domain, and let $\rho :\mathrm{\Omega }\to [-1,0)$ be a smooth strictly plurisubharmonic exhaustion function. We consider the logarithmic potential $g=-\log (-\rho )$ and the associated complete Kähler metric $\omega =d{d}^{c}g$. We prove that if $\rho$ satisfies the finite weighted Monge–Ampère mass condition ${\int }_{\mathrm{\Omega }}(-\rho {)}^{\epsilon }(d{d}^c\rho {)}^n<+\mathrm{\infty }$ for every $\epsilon >0$, then the bottom of the spectrum of the Laplace–Beltrami operator of $(\mathrm{\Omega },\omega )$ satisfies ${\lambda }_0({\mathrm{\Delta }}_{\omega },\mathrm{\Omega })=n^2$. The lower bound follows from the standard estimate applied to $g$, together with the inequality $\mathrm{\mid }\mathrm{\partial }g{\mathrm{\mid }}_{\omega }^2\le 1$. For the reverse inequality, for each $\alpha >n\mathrm{/}2$, we set $f=(-\rho {)}^{\alpha }$ and prove that $f\in W^{1,2}(\mathrm{\Omega },\omega )$ if and only if ${\int }_{\mathrm{\Omega }}(-\rho {)}^{2\alpha -n}(d{d}^c\rho {)}^n<+\mathrm{\infty }$. Under the finite weighted Monge–Ampère mass condition, this allows us to let $\alpha \downarrow n\mathrm{/}2$ in the Rayleigh quotient and obtain the upper bound ${\lambda }_0({\mathrm{\Delta }}_{\omega },\mathrm{\Omega })\le n^2$. As an application, Cegrell’s theorem gives a smooth strictly plurisubharmonic exhaustion with finite Monge–Ampère mass on every bounded hyperconvex domain; the associated complete Kähler metric constructed from this exhaustion therefore satisfies ${\lambda }_0({\mathrm{\Delta }}_{\omega },\mathrm{\Omega })=n^2$.