Prescribing negative curvature with cusps and conical singularities on compact surface

TL;DR

This paper proves the existence and uniqueness of conformal metrics with prescribed negative curvature on compact surfaces, allowing for cusps and conical singularities at finitely many points.

Contribution

It establishes a new result on prescribing negative curvature with singularities on compact surfaces, extending previous work to include cusps and conical points.

Findings

Existence of conformal metrics with prescribed negative curvature.

Uniqueness of such metrics under given conditions.

Inclusion of cusps and conical singularities in the prescribed curvature problem.

Abstract

On a compact surface, we prove existence and uniqueness of the conformal metric whose curvature is prescribed by a negative function away from finitely many points where the metric has prescribed angles presenting cusps or conical singularities.