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

Abstract

In this note, we study symmetry of solutions of the elliptic equation \begin{equation*} -Δ_{\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.

7 pages

References in corpus (2)

• The sphere covering inequality and its dual
• Non-axially symmetric solutions of a mean field equation on ${\mathbb{S}}^2$