Remarks on a mean field equation on $\mathbb{S}^{2}$
Analysis of PDEs
2019-05-28 v1 Differential Geometry
Abstract
In this note, we study symmetry of solutions of the elliptic equation \begin{equation*} -\Delta _{\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.
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Cite
@article{arxiv.1905.10842,
title = {Remarks on a mean field equation on $\mathbb{S}^{2}$},
author = {Changfeng Gui and Fengbo Hang and Amir Moradifam and Xiaodong Wang},
journal= {arXiv preprint arXiv:1905.10842},
year = {2019}
}
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7 pages