Remarks on a Mean Field Equation on $\mathbb{S}^2$

Authors

  • Changfeng Gui Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA
  • Fengbo Hang Courant Institute, New York University, New York, NY 10012, USA
  • Amir Moradifam Department of Mathematics, University of California at Riverside, Riverside, CA 92521, USA
  • Xiaodong Wang Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

DOI:

https://doi.org/10.4208/jms.v54n1.21.04

Keywords:

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

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.

Downloads

Published

2021-11-08

Issue

Vol. 54 No. 1 (2021)

Section

Articles