# Analysis Seminar

#### The Sphere Covering Inequality and Its Applications

Location: Warren Weaver Hall 1302

Date: Thursday, September 14, 2017, 11 a.m.

Synopsis:

We show that the total area of two distinct Gaussian
curvature 1 surfaces with the same conformal factor on the boundary,
which are also conformal to the Euclidean unit disk, must be at least
4π. In other words, the areas of these surfaces must cover the whole
unit sphere after a proper rearrangement. We refer to this lower bound
of total areas as the Sphere Covering Inequality. This inequality and
it’s generalizations are applied to a number of open problems
related to Moser-Trudinger type inequalities, mean field equations and
Onsager vortices, etc, and yield optimal results. In particular we
confirm the best constant of a Moser-Truidinger type inequality
conjectured by A. Chang and P. Yang in 1987. This is a joint work
Changfeng Gui.