Uniqueness of the Mean Field Equation and Rigidity of Hawking Mass

Abstract

In this paper, we prove that the even solution of the mean field equation $\Delta u=\lambda (1-e^u)$ on $S^2$ must be axially symmetric when $4<\lambda \le 8$. In particular, zero is the only even solution for $\lambda =6$. This implies the rigidity of Hawking mass for stable constant mean curvature(CMC) sphere with even symmetry.

arXiv comment15 pages

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Cited by in corpus (3)

• The sphere covering inequality and its dual
• Mean field equations on tori: existence and uniqueness of evenly symmetric blow-up solutions
• On Hawking mass and Bartnik mass of CMC surfaces