Symmetry and Rigidity Results for the Mean Field Equation and Hawking Mass on ( $\mathbb{S}$^2 )

In this paper, we establish symmetry results for solutions of the mean field equation $$\frac{\alpha }{2}\mathrm{\Delta }u+e^u-1=0$$ on ${\mathbb{S}}^2$ for $\frac{1}{3}\le \alpha <\frac{1}{2}$.The proofs utilize the Sphere Covering Inequality and incorporate topological arguments on ${\mathbb{S}}^2$. These results are further applied to demonstrate a rigidity property of the Hawking mass for stable constant mean curvature (CMC) spheres, addressing a question posed by Robert Bartnik in 2002. Our result unify and extend previous results on the rigidity of the Hawking mass for stable CMC spheres, encompassing earlier cases as special instances, specifically for surfaces that are not nearly spherical.