HCMU metrics with cusp singularities and conical singularities

TL;DR

This paper establishes necessary and sufficient conditions for the existence of HCMU metrics with both cusp and conical singularities on compact Riemann surfaces, advancing understanding of extremal Kähler metrics with singularities.

Contribution

It provides a complete characterization for the existence of HCMU metrics with mixed cusp and conical singularities, a problem previously not fully understood.

Findings

Derived necessary and sufficient conditions for existence

Characterized HCMU metrics with mixed singularities

Enhanced understanding of extremal Kähler metrics

Abstract

An HCMU metric is a conformal metric which has a finite number of singularities on a compact Riemann surface and satisfies the equation of the extremal K\"{a}hler metric. In this paper, we give a necessary and sufficient condition for the existence of a kind of HCMU metrics which has both cusp singularities and conical singularities.