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PreprintCC BY 4.0arXiv

Symmetry and Rigidity Results for the Mean Field Equation and Hawking Mass on 𝕊2

  • Changfeng Gui
  • Amir Moradifam
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Abstract

In this paper, we establish symmetry results for solutions of the mean field equation α/2 Δu + eu − 1 = 0 on $ \mathbb{S}^2 $ for 1/3≤ α≤ 1, under a geometric condition (ℋ) introduced below. The proofs utilize the Sphere Covering Inequality and incorporate topological arguments on $ \mathbb{S}^2 $. These results are further applied to demonstrate a rigidity property of the Hawking mass for stable constant mean curvature (CMC) spheres, addressing a question posed by Robert Bartnik in 2002. Our results unify and extend previous rigidity results obtained under symmetry or hemispherical balance assumptions, and apply beyond the nearly spherical setting.

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Cite

BibTeXRIS
BibTeX
@misc{gui2026symmetry,
  title = {{Symmetry and Rigidity Results for the Mean Field Equation and Hawking Mass on $\mathbbS^2$}},
  author = {Changfeng Gui and Amir Moradifam},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.15448},
  eprint = {2605.15448},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.15448},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Symmetry and Rigidity Results for the Mean Field Equation and Hawking Mass on 𝕊^2
AU  - Changfeng Gui
AU  - Amir Moradifam
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2605.15448
UR  - https://arxiv.org/abs/2605.15448
AB  - In this paper, we establish symmetry results for solutions of the mean field equation α/2 Δu + e^u − 1 = 0 on $ \mathbb{S}^2 $ for 1/3≤ α≤ 1, under a geometric condition (ℋ) introduced below. The proofs utilize the Sphere Covering Inequality and incorporate topological arguments on $ \mathbb{S}^2 $. These results are further applied to demonstrate a rigidity property of the Hawking mass for stable constant mean curvature (CMC) spheres, addressing a question posed by Robert Bartnik in 2002. Our results unify and extend previous rigidity results obtained under symmetry or hemispherical balance assumptions, and apply beyond the nearly spherical setting.
ER  -