UBC DG MP PDE Seminar: Amir Moradifam

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 ℝ3. This is achieved by showing that all solutions to the mean field equation α2Δu+eu−1=0 on 𝕊2 are axially symmetric when 13≤α≤1. The proof leverages the Sphere Covering Inequality and employs topological arguments on 𝕊2, enabling us to establish symmetry results for α≥13, 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 α≥12. This is a joint work with Changdeng Gui.