# THE SPHERE COVERING INEQUALITY AND ITS APPLICATIONS

In this article  we introduce a new geometric inequality:  the Sphere Covering Inequality,  which  states that   the total area  of two distinct  surfaces conformal to an Euclidean disk with Gaussian curvature less than 1 and  the same conformal factor on the boundary,  must be at least $4 \pi$.  In other words,  the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We apply the Sphere Covering Inequality to show the best constant of a Moser-Trudinger type inequality.    Other applications of this inequality include the classification of certain Onsager vortices  on the sphere,  the radially symmetry of solutions to Gaussian curvature equation on the plane, classification of solutions for mean field equations on flat tori and  the standard sphere, etc., and lead to the  resolution of several open problems in these areas.   The article is a joint work with Amir Moradifam from UC Riverside
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