A gluing construction of constant scalar curvature K\"ahler metrics of Poincar\'e type

TL;DR

This paper develops a gluing method to construct constant scalar curvature K"ahler metrics of Poincaré type on non-compact manifolds obtained by removing finitely many points from a compact K"ahler manifold with no non-trivial holomorphic vector fields, assuming the original admits such a metric.

Contribution

It introduces a novel gluing construction for constant scalar curvature K"ahler metrics of Poincaré type on punctured manifolds, expanding the class of known solutions.

Findings

Existence of Poincaré type CSC K"ahler metrics on punctured manifolds

Extension of CSC K"ahler metrics to non-compact settings

New techniques for gluing K"ahler metrics of Poincaré type

Abstract

Given a compact K\"ahler manifold with no non-trivial holomorphic vector field, assume it admits a constant scalar curvature K\"ahler metric. Fix finitely many points, we show the existence of constant scalar curvature K\"ahler metrics of Poincar\'e type on the complement of these points in the compact manifold.