Xuezhang Chen, Shihong Zhang
Published 2023-09-09Version 1
We establish Moser-Trudinger-Onofri inequalities under constraint of a deviation of the second order moments from $0$, which serves as an intermediate one between Chang-Hang's inequalities under first and second order moments constraints. A threshold for the deviation is a uniqueness criterion for the mean field equation $$-a\Delta_{\mathbb{S}^2}u+1=e^{2u} \quad \mathrm{~~on~~} \quad \mathbb{S}^2$$ when the constant $a$ is close to $\frac{1}{2}$.
Remarks on a mean field equation on $\mathbb{S}^{2}$
Uniqueness of the Mean Field Equation and Rigidity of Hawking Mass
Mean field equations on a closed Riemannian surface with the action of an isometric group
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