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

Abstract: Let $Ω\subset\mathbb{C}<sup>{n}$ 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 $ω=dd<sup>{c}g$. We prove that if $ρ$ satisfies the finite weighted Monge--Ampère mass condition $\int_Ω(-ρ)<sup>{\varepsilon}(dd<sup>{c}ρ)<sup>{n}&lt;+\infty$ for every $\varepsilon&gt;0$, then the bottom of the spectrum of the Laplace--Beltrami operator of $(Ω,ω)$ satisfies $λ<em>{0}(Δ</em>ω,Ω)=n<sup>{2}$. The lower bound follows from the standard estimate applied to $g$, together with the inequality $|\partial g|<em>ω^{2}\le 1$. For the reverse inequality, for each $α&gt;n/2$, we set $f=(-ρ)<sup>α$ and prove that $f\in W<sup>{1,2}(Ω,ω)$ if and only if $\int</em>Ω(-ρ)<sup>{2α-n}(dd<sup>{c}ρ)<sup>{n}&lt;+\infty$. 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 $λ<em>{0}(Δ</em>ω,Ω)\le n<sup>{2}$. 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 $λ<em>{0}(Δ</em>ω,Ω)=n<sup>{2}$.

• The paper proves that the spectral bottom of complete Kähler metrics equals n² under finite weighted Monge–Ampère mass conditions.
• It leverages matrix norm estimates and explicit Sobolev characterizations with test functions to rigorously derive the Laplace–Beltrami bounds.
• The results broaden complex geometric analysis by relaxing boundary regularity and ensuring applicability to bounded hyperconvex domains.

Spectrum Lower Bound for Complete Kähler Metrics via Finite-Mass Plurisubharmonic Exhaustions

Introduction and Context

The paper "Bottom of the Spectrum of Complete Kähler Metrics from Finite-Mass Plurisubharmonic Exhaustions" (2607.03036) advances the spectral analysis of complete Kähler metrics constructed from plurisubharmonic (psh) exhaustion functions using finite weighted Monge–Ampère mass conditions. Specifically, it characterizes the infimum (bottom) of the spectrum $\lambda_0$ of the Laplace–Beltrami operator for a class of such Kähler metrics on bounded domains in $\mathbb{C}^n$.

The principal result generalizes earlier computations of the spectrum for metrics arising from boundary-regular defining functions, as seen in the work of Li–Tran, to the context of exhaustions defined only in the interior, relaxing boundary regularity in favor of integrability conditions on the Monge–Ampère mass.

Main Results

Let $\Omega\subset\mathbb{C}^{n}$ be a bounded domain, and let $\rho:\Omega\to[-1,0)$ be a smooth strictly psh exhaustion. The Kähler metric is constructed as $\omega=dd^c(-\log(-\rho))$. The principal theorem establishes:

If $$\int_{\Omega} (-\rho)^\varepsilon (dd^c \rho)^n < \infty$$ for all $\varepsilon>0$, then the bottom of the spectrum $\lambda_0(\Delta_\omega,\Omega)$ of the metric satisfies $\lambda_0 = n^2$.

This condition is referred to as the finite weighted Monge–Ampère mass condition. In the case that $\rho$ has finite Monge–Ampère mass—i.e., $\mathbb{C}^n$0—the result applies due to the monotonicity of $\mathbb{C}^n$1 for $\mathbb{C}^n$2.

A corollary addresses the hyperconvex case: any bounded hyperconvex domain $\mathbb{C}^n$3 admits such a psh exhaustion (by Cegrell's theorem), hence the constructed complete Kähler metric satisfies $\mathbb{C}^n$4.

Technical Approach

The derivation consists of two central parts:

1. Lower Bound Construction: Utilizing a matrix norm estimate, the paper employs an inequality of Li–Tran, namely that if the squared norm of the gradient of the potential ($\mathbb{C}^n$5) with respect to the Kähler metric is at most one, then $\mathbb{C}^n$6. The paper provides a direct computation that $\mathbb{C}^n$7 holds for this metric structure.
2. Upper Bound via Test Functions: For the upper bound, the authors replace boundary regularity assumptions with integrability of suitable test functions. Letting $\mathbb{C}^n$8 for $\mathbb{C}^n$9 and exploiting explicit calculations of their $\Omega\subset\mathbb{C}^{n}$0-norms, the paper establishes equivalence between membership in the energy space and a weighted mass integral:

$\Omega\subset\mathbb{C}^{n}$1

The Rayleigh quotient for these test functions then yields exact values, and taking the limit $\Omega\subset\mathbb{C}^{n}$2 delivers the bound $\Omega\subset\mathbb{C}^{n}$3.

The principal analytical innovations are the explicit Sobolev characterizations for test function admissibility and rigorous control of boundary contributions via precise Monge–Ampère integrability.

Implications and Applications

From a geometric perspective, the result anchors the spectral gap at $\Omega\subset\mathbb{C}^{n}$4 for a broad class of complete Kähler metrics associated with finite-mass exhaustions, encompassing all bounded hyperconvex domains. This is optimal in the prevailing context, matching the lower bounds established under stronger curvature requirements in classical work.

The absence of boundary regularity conditions in favor of finite weighted Monge–Ampère mass substantially broadens the applicability; any geometry permitting the existence of such an exhaustion inherits the spectral lower bound. The construction is robust under rescaling and composition with strictly increasing transformations, highlighting the analytic flexibility of the exhaustion approach. The result also motivates refined questions regarding the Kähler–Einstein asymptotics and the behavior of the Ricci form, as the paper notes that the prescribed construction does not, in general, deliver asymptotic Kähler–Einstein geometry without further assumptions on the exhaustion function beyond finite mass.

In the context of complex geometry and PDEs, this establishes a spectral foundation for potential-theoretic arguments and supports the analysis of heat kernel behavior, the study of $\Omega\subset\mathbb{C}^{n}$5 cohomology, and the investigation of geometric flows on noncompact Kähler manifolds with singular or weak boundary regularity. The methodology also sets the stage for generalizations to more abstract settings such as hyperconvex complex manifolds and $\Omega\subset\mathbb{C}^{n}$6-hyperconvex domains.

Conclusion

The paper rigorously determines that the bottom of the spectrum for complete Kähler metrics defined by finite-mass strictly plurisubharmonic exhaustions is precisely $\Omega\subset\mathbb{C}^{n}$7 under general integrability conditions. This generalizes and strengthens previous results by circumventing rigidity in boundary regularity, establishing a direct Monge–Ampère mass criterion. The work both elucidates the interplay between Monge–Ampère measure and spectral theory and invites further exploration of exhaustion potentials that reconcile spectral optimality with enhanced curvature control.