Extremal K\"ahler metrics on blow-ups of parabolic ruled surfaces

TL;DR

This paper constructs new extremal K"ahler metrics on blow-ups of parabolic ruled surfaces using a gluing method, expanding the class of known extremal metrics with non-constant scalar curvature.

Contribution

It introduces a novel application of the Arezzo-Pacard-Singer gluing technique to parabolic ruled surfaces, producing extremal K"ahler metrics on their blow-ups.

Findings

Existence of extremal K"ahler metrics on specific blow-ups

Application of gluing construction to parabolic structures

Extension of extremal metric examples to new geometric settings

Abstract

New examples of extremal K\"ahler metrics on blow-ups of parabolic ruled surfaces are constructed. The method is based on the gluing construction of Arezzo, Pacard and Singer. This enables to endow ruled surfaces of the form $\mathbb{P}(\mathcal{O}\oplus L)$ with special parabolic structures such that the associated iterated blow-up admits an extremal metric of non-constant scalar curvature.