Uniqueness of the mean field equation and rigidity of Hawking Mass

  • Yuguang Shi
  • Jiacheng Sun
  • Gang Tian
  • Dongyi WeiEmail author


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 sphere with even symmetry.

Mathematics Subject Classification

35J20 53C21 



We are very grateful to Juncheng Wei for suggesting us the sphere covering inequality of Gui and Moradifam [11]. The second author also want to thank Jie Qing for pointing out the relation between rigidity of Hawking mass and eigenvalue problem. Many thanks to Changfeng Gui for many discussions on mean field equation.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Yuguang Shi
    • 1
  • Jiacheng Sun
    • 1
  • Gang Tian
    • 1
  • Dongyi Wei
    • 1
    Email author
  1. 1.School of Mathematical Sciences and BICMRPeking UniversityBeijingPeople’s Republic of China

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