Prescribing positive curvature with conical singularities on $\mathbb S^2$

TL;DR

This paper studies conformal metrics with positive curvature and conical singularities on the sphere, proving convergence results and criteria for nonexistence by analyzing bubble trees and area identities.

Contribution

It introduces a convergence theorem for such metrics and provides a new nonexistence criterion based on detailed bubble tree analysis.

Findings

Established a convergence theorem for conformal metrics with conical singularities.

Derived a criterion for nonexistence of solutions in certain prescribing data regions.

Performed a detailed analysis of bubble trees and area identities during convergence.

Abstract

For conformal metrics with conical singularities and positive curvature on $\mathbb S^2$, we prove a convergence theorem and apply it to obtain a criterion for nonexistence in an open region of the prescribing data. The core of our study is a fine analysis of the bubble trees and an area identity in the convergence process.