| Abstract: |
| We demonstrate that the total area of two distinct surfaces with Gaussian curvature 1, which share the same conformal factor on the boundary and are conformal to the Euclidean unit disk, must be at least $4\pi$. In other words, the areas of these surfaces must cover the entire unit sphere after an appropriate rearrangement. We refer to this minimum total area as the Sphere Covering Inequality. This inequality and its generalizations are applied to several open problems related to Moser-Trudinger type inequalities, mean field equations, and Onsager vortices, among others, yielding optimal results. In particular, we confirm the best constant of a Moser-Trudinger type inequality that was conjectured by A. Chang and P. Yang in 1987. This work is a collaboration with Changfeng Gui. |
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