Abstract
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.
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References
Bandle, C.: Isoperimetric Inequalities and Applications, Volume 7 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, Mass.-London, (1980)
Bartnik, R.: Mass and 3-metrics of non-negative scalar curvature. In: Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 231–240, Higher Ed. Press, Beijing (2002)
Bartolucci, D., Lin, C.S.: Existence and uniqueness for mean field equations on multiply connected domains at the critical parameter. Math. Ann. 359, 1–44 (2014)
Bol, G.: Isoperimetrische Ungleichungen \(f\ddot{u}r\) Bereiche auf \(Fl\ddot{a}chen\). Jber. Deutsch. Math. Verein. 51, 219–257 (1941)
Bray, H.L.: The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature. Thesis, Stanford University, Stanford (1997)
Chen, C.C., Lin, C.S.: Topological degree for a mean field equation on Riemann surfaces. Commun. Pure Appl. Math. 56(12), 1667–1727 (2003)
Christodoulou, D., Yau, S.-T.: Some remarks on the quasi-local mass. Contemp. Math. 71, 9–14 (1988)
Ding, W., Jost, J., Li, J., Wang, G.: The differential equation \(\Delta u=8\pi -8\pi he^u \) on a compact Riemann surface. Asian J. Math. 1, 230–248 (1997)
Ding, W., Jost, J., Li, J., Wang, G.: An analysis of the two-vortex case in the Chern-Simons Higgs model. Calc. Var. 7, 87–97 (1998)
Ghoussoub, N., Lin, C.S.: On the best constant in the Moser-Onofri-Aubin inequality. Commun. Math. Phys. 298(3), 869–878 (2010)
Gui, C., Moradifam, A.: The Sphere Covering Inequality and Its Applications, arXiv:1605.06481v3. Invent. math. (2018). https://doi.org/10.1007/s00222-018-0820-2
Gui, C., Moradifam, A.: Uniqueness of Solutions of Mean Field Equations in \(\mathbb{R}^2\), arXiv:1612.08403v2. Proc. Amer. Math. Soc. 146(3), 1231–1242 (2018)
Hang, F., Wang, X.: Rigidity and non-rigidity results on the sphere. Commun. Anal. Geom. 14, 91–106 (2006)
Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose Inequality. J. Differ. Geom. 59, 353–437 (2001)
Ji, D., Shi, Y., Zhu, B.: Exhaustion of isoperimetric regions in asymptotically hyperbolic manifolds with scalar curvature \(R\ge -6\), preprint, arXiv:1512.02732v2, Commun. Anal. Geom. 26(3), 627–658 (2018)
Kazdan, J., Warner, F.: Curvature functions for compact 2-manifolds. Ann. Math. 99, 14–47 (1974)
Li, Y.Y.: Harnack type inequality: the method of moving planes. Commun. Math. Phys. 200, 421–444 (1999)
Lin, C.S.: Topological degree for mean field equations on \(S^2\). Duke Math. J. 104(3), 501–536 (2000)
Lin, C.S., Lucia, M.: One-dimensional symmetry of periodic minimizers for a mean field equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6(2), 269–290 (2007)
Miao, P.: Positive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys. 6, 1163–1182 (2002)
Onofri, E.: On the positivity of the effective action in a theory of random surfaces. Commun. Math. Phys. 86, 321–326 (1982)
Shi, Y., Tam, L.F.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Differ. Geom. 62, 79–125 (2002)
Shi, Y., Tam, L.F.: Rigidity of compact manifolds and positivity of quasi-local mass. Class. Quantum Gravity 24(9), 2357–2366 (2007)
Shi, Y.: The isoperimetric inequality on asymptotically flat manifolds with nonnegative scalar curvature. Int. Math. Res. Not. IMRN 22, 7038–7050 (2016)
Sun, J.: Rigidity of Hawking mass for surfaces in three manifolds. arXiv:1703.02352, Pacific J. Math. 292(2), 479–504 (2018)
Suzuki, T.: Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire 9(4), 367–397 (1992)
Acknowledgements
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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Shi, Y., Sun, J., Tian, G. et al. Uniqueness of the mean field equation and rigidity of Hawking Mass. Calc. Var. 58, 41 (2019). https://doi.org/10.1007/s00526-019-1496-1
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DOI: https://doi.org/10.1007/s00526-019-1496-1