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Volume 54, Issue 1
Remarks on a Mean Field Equation on $\mathbb{S}^2$

Changfeng Gui, Fengbo Hang, Amir Moradifam & Xiaodong Wang

DOI: 10.4208/jms.v54n1.21.04

J. Math. Study, 54 (2021), pp. 81-88.

Published online: 2021-01

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  • 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 consideration 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.

  • Keywords

Semilinear elliptic equation, sphere covering inequality, rigidity of Hawking mass.

  • AMS Subject Headings

35J61, 58J05, 35B06

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

changfeng.gui@utsa.edu (Changfeng Gui)

fengbo@cims.nyu.edu (Fengbo Hang)

moradifam@math.ucr.edu (Amir Moradifam)

xwang@math.msu.edu (Xiaodong Wang)

  • BibTex
  • RIS
  • TXT
@Article{JMS-54-81, author = {Gui , ChangfengHang , FengboMoradifam , Amir and Wang , Xiaodong}, title = {Remarks on a Mean Field Equation on $\mathbb{S}^2$}, journal = {Journal of Mathematical Study}, year = {2021}, volume = {54}, number = {1}, pages = {81--88}, 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 consideration 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.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v54n1.21.04}, url = {http://global-sci.org/intro/article_detail/jms/18599.html} }
TY - JOUR T1 - Remarks on a Mean Field Equation on $\mathbb{S}^2$ AU - Gui , Changfeng AU - Hang , Fengbo AU - Moradifam , Amir AU - Wang , Xiaodong JO - Journal of Mathematical Study VL - 1 SP - 81 EP - 88 PY - 2021 DA - 2021/01 SN - 54 DO - http://doi.org/10.4208/jms.v54n1.21.04 UR - https://global-sci.org/intro/article_detail/jms/18599.html KW - Semilinear elliptic equation, sphere covering inequality, rigidity of Hawking mass. AB -

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 consideration 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.

Gui , ChangfengHang , FengboMoradifam , Amir and Wang , Xiaodong. (2021). Remarks on a Mean Field Equation on $\mathbb{S}^2$. Journal of Mathematical Study. 54 (1). 81-88. doi:10.4208/jms.v54n1.21.04
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