Remarks on a mean field equation on $\mathbb{S}^{2}$

TL;DR

This paper investigates symmetry properties of solutions to a specific elliptic equation on the 2-sphere, with implications for the rigidity of Hawking mass in general relativity, identifying conditions for solutions to be constant.

Contribution

It provides new conditions ensuring solutions are constant, advancing understanding of the rigidity of Hawking mass in the context of stable CMC spheres.

Findings

Certain conditions guarantee solutions are constant

Implications for rigidity of Hawking mass in general relativity

Enhances understanding of symmetry in elliptic equations on spheres

Abstract

In this note, we study symmetry of solutions of the elliptic equation \begin{equation*} -\Delta _{\mathbb{S}^{2}}u+3=e^{2u}\ \ \hbox{on}\ \ \mathbb{S}^{2}, \end{equation*} that arises in the study of rigidity problem of Hawking mass in general relativity. We provide various conditions under which this equation has only constant solutions, and consequently imply the rigidity of Hawking mass for stable constant mean curvature (CMC) sphere.