Classification of non-CSC extremal K\"{a}hler metrics on K-surfaces $S^2_{\{\alpha\}}$ and $S^2_{\{\alpha,\beta\}}$

TL;DR

This paper classifies specific non-constant scalar curvature extremal Kähler metrics with singularities on certain K-surfaces, expanding understanding of their geometric structure.

Contribution

It provides a classification of non-CSC HCMU metrics on K-surfaces with particular singularity configurations, a novel contribution to Kähler geometry.

Findings

Classification of non-CSC HCMU metrics on $S^2_{\\\\{\\alpha\\\}$ and $S^2_{\\\\\{\\alpha,\eta\\\}}$

Identification of geometric properties of these metrics

Extension of known results on extremal Kähler metrics

Abstract

We commonly refer to an extremal K\"{a}hler metric with finitely many singularities on a compact Riemann surface as an HCMU (Hessian of the Curvature of the Metric is Umbilical) metric. In this study, we specifically classify non-CSC HCMU metrics on the K-surfaces $S^2_{\{\alpha\}}$ and $S^2_{\{\alpha,\beta\}}$.