$\sigma_2$ Yamabe problem on conic 4-spheres

TL;DR

This paper investigates the existence and uniqueness of constant $\sigma_2$ metrics on conic 4-spheres, establishing necessary conditions and characterizing the boundary of the moduli space with symmetric conic spheres.

Contribution

It provides a necessary condition for the existence of constant $\sigma_2$ metrics on conic 4-spheres and describes the boundary of the moduli space with symmetric conic spheres.

Findings

Necessary condition for existence of constant $\sigma_2$ metrics.

Uniqueness result for sharp conditions, similar to Troyanov's in 2D.

Boundary of moduli space characterized by symmetric conic spheres with 2 conic points.

Abstract

We discuss the constant $\sigma_{2}$ problem for conic 4-spheres. Based on earlier works of Chang-Han-Yang and Han-Li-Teixeira, we are able to find a necessary condition for the existence problem. In particular, when the condition is sharp, we have the uniqueness result similar to that of Troyanov in dimension 2. It indicates that the boundary of the moduli of all conic 4-spheres with constant $\sigma_{2}$ metrics consists of conic spheres with 2 conic points and rotational symmetry.