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

Abstract

In this paper, we establish symmetry results for solutions of the mean field equation \[ \fracα{2} Δu + e^u - 1 = 0 \] on ${\mathbb{S}}^2$ for $\frac{1}{3}\le \alpha \le 1$, under a geometric condition $(\mathcal{H})$ introduced below. 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 results unify and extend previous rigidity results obtained under symmetry or hemispherical balance assumptions, and apply beyond the nearly spherical setting.