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

Abstract

Let $Ω\subset\mathbb{C}^{n}$ be a bounded domain, and let $ρ:Ω\to[-1,0)$ be a smooth strictly plurisubharmonic exhaustion function. We consider the logarithmic potential $g=-\log(-ρ)$ and the associated complete Kähler metric $ω=dd^{c}g$. We prove that if $ρ$ satisfies the finite weighted Monge--Ampère mass condition $\int_Ω(-ρ)^{\varepsilon}(dd^{c}ρ)^{n}<+\infty$ for every $\varepsilon>0$, then the bottom of the spectrum of the Laplace--Beltrami operator of $(Ω,ω)$ satisfies $λ_{0}(Δ_ω,Ω)=n^{2}$. The lower bound follows from the standard estimate applied to $g$, together with the inequality $|\partial g|_ω^{2}\le 1$. For the reverse inequality, for each $α>n/2$, we set $f=(-ρ)^α$ and prove that $f\in W^{1,2}(Ω,ω)$ if and only if $\int_Ω(-ρ)^{2α-n}(dd^{c}ρ)^{n}<+\infty$. Under the finite weighted Monge--Ampère mass condition, this allows us to let $α\downarrow n/2$ in the Rayleigh quotient and obtain the upper bound $λ_{0}(Δ_ω,Ω)\le n^{2}$. 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 $λ_{0}(Δ_ω,Ω)=n^{2}$.