Asymptotic expansions of complete K\"ahler-Einstein metrics with finite volume on quasi-projective manifolds

TL;DR

This paper provides an elementary proof for the asymptotic expansion of complete K"ahler-Einstein metrics with finite volume on quasi-projective manifolds, using ODE solutions and spectral theory.

Contribution

It offers a simplified proof of existing asymptotic expansion formulas for K"ahler-Einstein metrics on quasi-projective manifolds.

Findings

Elementary proof of asymptotic expansion formula

Application of spectral theory and ODE solutions

Clarification of the behavior of metrics near divisors

Abstract

We give an elementary proof to the asymptotic expansion formula of Rochon-Zhang for the unique complete K\"ahler-Einstein metric of Cheng-Yau, Kobayashi, Tian-Yau and Bando on quasi-projective manifolds. The main tools are the solution formula for second order ODE's with constant coefficients and spectral theory for Laplacian operator on a closed manifold.