Frontiers in Mathematical Sciences - 8th Conference


TITLE  
The Sphere Covering Inequality and Its Applications


SPEAKER  
Amir Moradifam
University of California, Riverside





ABSTRACT

I will talk about a new geometric inequality: the Sphere Covering Inequality. The inequality states that the total area of two 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 4π. 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 solve several open problems about uniqueness and symmetry of solutions of mean field type equations. In particular we apply this inequality to prove an old conjecture of A. Chang and P. Yang about the best constant of a Moser-Trudinger type inequality.