Bottom of the Spectrum of Complete Kähler 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 Kähler metric $\omega =d{d}^{c}g$. We prove that if $\rho$ satisfies the finite weighted Monge--Ampère mass condition ${\int }_{\Omega }(-\rho {)}^{\epsilon }(d{d}^c\rho {)}^n<+\infty$ for every $\epsilon >0$, 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 ${\int }_{\Omega }(-\rho {)}^{2\alpha -n}(d{d}^c\rho {)}^n<+\infty$. Under the finite weighted Monge--Ampère 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--Ampère mass on every bounded hyperconvex domain; the associated complete Kähler metric constructed from this exhaustion therefore satisfies ${\lambda }_0({\Delta }_{\omega },\Omega )=n^2$.

12 pages, no figures