The sphere covering inequality and its applications (Q1628415)

Let \(\Omega\) be a \(C^2\) simply connected region in \(\mathbb{R}^2\). Consider the equation \(\Delta v+e^{2v}=0\) on \(\Omega\). The conformally equivalent metric \(dg=e^{2v}dy\) has constant Gaussian curvature \(+1\) and total area \(\int_{\Omega}e^{2v}dy\). Suppose there is another surface \((\Omega,\tilde g)\) with a distinct conformally metric such that \(\tilde g=g\) on the boundary of \(\Omega\), then the authors establish the inequality \(\tilde A+A\geq4\pi\). A similar inequality is established in a more general setting that is used to establish the best constant in a Moser-Trudinger-type inequality, mean field equations, Onsager vortices, and other results.