Spectrum of Complete Kähler Metrics on Bounded Hyperconvex Domains
Abstract
Let Ω⊂ℂn be a bounded domain, and let ρ:Ω→[−1,0) be a smooth strictly plurisubharmonic exhaustion function. We consider the logarithmic potential g=−log(−ρ) and the associated complete Kähler metric ω=ddc g. We prove that if ρ satisfies the finite weighted Monge-Ampère mass condition with weight (−ρ)ε for every ε>0, then the spectrum and the essential spectrum of the Laplace-Beltrami operator of (Ω,ω) both coincide with [n2,∞). In particular, the bottom of the spectrum satisfies λ0(Δω,Ω)=n2, and n2 is not an L2-eigenvalue. For each α> n/2, we set f=(−ρ)α and prove that f belongs to W1,2(Ω,ω) if and only if the Monge-Ampère mass of ρ weighted by (−ρ)2α−n is finite. The integral identities underlying this equivalence, applied to the complex powers (−ρ)α+it, give, after normalization, a singular Weyl sequence at every n2+4t2, t∈ℝ, as α↓n/2. Combining our theorem with Cegrell's finite-mass exhaustion theorem, we obtain a complete Kähler metric with spectrum and essential spectrum equal to [n2,∞) on every bounded hyperconvex domain.
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BibTeX
@misc{choi2026spectrum,
title = {{Spectrum of Complete Kähler Metrics on Bounded Hyperconvex Domains}},
author = {Young-Jun Choi and Jiwon Brandon Jeong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.03036},
eprint = {2607.03036},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.03036},
}
RIS
TY - UNPB
TI - Spectrum of Complete Kähler Metrics on Bounded Hyperconvex Domains
AU - Young-Jun Choi
AU - Jiwon Brandon Jeong
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2607.03036
UR - https://arxiv.org/abs/2607.03036
AB - Let Ω⊂ℂ^n be a bounded domain, and let ρ:Ω→[−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 with weight (−ρ)^ε for every ε>0, then the spectrum and the essential spectrum of the Laplace-Beltrami operator of (Ω,ω) both coincide with [n^2,∞). In particular, the bottom of the spectrum satisfies λ_0(Δ_ω,Ω)=n^2, and n^2 is not an L^2-eigenvalue. For each α> n/2, we set f=(−ρ)^α and prove that f belongs to W^1,2(Ω,ω) if and only if the Monge-Ampère mass of ρ weighted by (−ρ)^2α−n is finite. The integral identities underlying this equivalence, applied to the complex powers (−ρ)^α+it, give, after normalization, a singular Weyl sequence at every n^2+4t^2, t∈ℝ, as α↓n/2. Combining our theorem with Cegrell's finite-mass exhaustion theorem, we obtain a complete Kähler metric with spectrum and essential spectrum equal to [n^2,∞) on every bounded hyperconvex domain.
ER -