An improved Moser-Aubin-Onofri inequality for axially symmetric functions on \(S^2\)
Let \(J_\alpha\) be the functional defined on the Sobolev space \(H^{1,2} (S^2)\) by \[ J_\alpha (g)= \alpha \int_{S^2} | \nabla g|^2 d\omega +2\int_{S^2} gd\omega-\ln\int_{S^2} e^{2g} d\omega. \] It is known that by restricting \(J_\alpha\) to the class \({\mathcal G}\) of functions \(g\) for which \(e^{2g}\) has center of mass equal to zero, i.e., \(\int_{S^2} e^{2g} xd\omega =0\), then for any \(\alpha> 1/2\), \(J_\alpha\) is bounded below. This led \textit{S.-Y. A. Chang} and \textit{P. C. Yang} [J. Differ. Geom. 27, 259-296 (1988; Zbl 0649.53022)] to the conjecture: (1) If \(\alpha \geq 1/2 \), then \(\inf_{g\in {\mathcal G}_r} J_\alpha (g)=0\). (2) If \(\alpha <1/2\), then \(J_\alpha\) is unbounded below. This paper deals with this conjecture in the case of axially symmetric functions, presenting the following Theorem. For the set \({\mathcal G}_r\) of all functions \(g\) on \(H^1 (-1,1)\) for which \(\int^1_{-1} e^{2g(x)} xdx=0\), (1) if \(\alpha> 16/25- \varepsilon\) for small enough \(\varepsilon\), then \(\inf_{g\in {\mathcal G}_r} J_\alpha (g)=0\). (2) If \(\alpha<1/2\), the functional \(J_\alpha\) is unbounded below.
scientific article; zbMATH DE number 1140139