Expansions of solutions to extremal metric type equations on blow-ups of cscK surfaces

TL;DR

This paper investigates the detailed expansions of solutions to extremal metric equations on blow-ups of cscK surfaces, advancing understanding of K-stability and metric existence in complex geometry.

Contribution

It provides new asymptotic expansions of extremal metric solutions on blow-ups, linking stability conditions to metric existence on Kähler surfaces.

Findings

Derived explicit solution expansions near blow-up points.

Established connections between K-stability and extremal metrics.

Enhanced understanding of metric behavior on blow-up surfaces.

Abstract

The aim of this article is to study expansions of solutions to an extremal metric type equation on the blow-up of constant scalar curvature K\"ahler surfaces. This is related to finding constant scalar curvature K\"ahler (cscK) metrics on K-stable blow-ups of extremal K\"ahler surfaces