Rigidity of Hawking Mass for Stable Constant Mean Curvature Spheres

In this talk, I will discuss recent work in which we establish the rigidity of the Hawking mass for stable constant mean curvature spheres, addressing a problem posed by Robert Bartnik in 2002. More precisely, we demonstrate that any complete Riemannian three-manifold with non-negative scalar curvature, whose boundary is a stable constant mean curvature sphere with zero Hawking mass, must be isometric to a Euclidean ball in \( \mathbb{R}^3 \). This is achieved by showing that all solutions to the mean field equation \( \frac{\alpha}{2}\Delta u + e^u - 1 = 0 \) on \( \mathbb{S}^2 \) are axially symmetric when \( \frac{1}{3} \leq \alpha \leq 1 \). The proof leverages the Sphere Covering Inequality and employs topological arguments on $\mathbb{S}^2$, enabling us to establish symmetry results for $\alpha \geq \frac{1}{3}$, a significant improvement over the results in [C. Gui and A. Moradifam. The sphere covering inequality and its applications. Invent. Math., 214(3):1169-1204, 2018], which were applicable only for $\alpha \geq \frac{1}{2}$. This is a joint work with Changdeng Gui.