SEMINARS
The Sphere Covering Inequality and its applications

2018-08-10　16:00 — 17:00

Middle Lecture Room

Changfeng Gui

University of Texas at San Antonio and Central South University

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.