Asymptotic properties of extremal K\"ahler metrics of Poincar\'e type

Abstract

Consider a compact K\"ahler manifold X with a simple normal crossing divisor D, and define Poincar\'e type metrics on X\D as K\"ahler metrics on X\D with cusp singularities along D. We prove that the existence of a constant scalar curvature (resp. an extremal) Poincar\'e type K\"ahler metric on X\D implies the existence of a constant scalar curvature (resp. an extremal) K\"ahler metric, possibly of Poincar\'e type, on every component of D. We also show that when the divisor is smooth, the constant scalar curvature/extremal metric on X\D is asymptotically a product near the divisor.