Asymptotics of small eigenvalues on degenerations of K\"ahler manifolds

TL;DR

This paper determines the precise asymptotic behavior of small eigenvalues of the Laplacian during degenerations of compact K"ahler manifolds, extending prior results to higher dimensions.

Contribution

It generalizes recent results on eigenvalue asymptotics to higher-dimensional degenerations using advanced inequalities and Monge-Ampère equation techniques.

Findings

Derived exact asymptotic rates of small eigenvalues during degenerations.

Extended previous results to higher-dimensional K"ahler manifolds.

Provided estimates for degenerations with reducible singular fibers.

Abstract

We derive the exact asymptotic rates of the small eigenvalues of the Laplacian on one-parameter degenerations of compact K\"ahler manifolds equipped with induced background metrics. This generalizes a recent result of Dai and Yoshikawa to higher dimensions. To achieve this, we combine Li's uniform Skoda inequality with the method of auxiliary Monge-Amp\`ere equations, introduced by Guo--Phong--Song--Sturm--Tong and adapted by Guedj--T\^o. As an application, we establish estimates for degenerations of compact K\"ahler manifolds with reducible singular fibers.