The Sphere Covering Inequality and its application to a Moser-Trudinger type inequality and mean field equations

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Changfeng Gui
Tue, 03/14/2017 - 3:00pm - 4:00pm
Song-Ying Li

In this talk, I will introduce a new geometric inequality:  the Sphere Covering Inequality. The inequality  states that   the  total area  of two {\it distinct}  surfaces with Gaussian curvature  less than 1,   which  are also conformal to  the Euclidean unit disk  with 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 conjectured by A. Chang and P. Yang.    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.   The resolution of several open problems in these areas will  be presented.  The talk is based on joint work with Amir Moradifam from UC Riverside.