The sphere covering inequality and its applications

In this paper, we showthat the total area of two distinct surfaces with Gaussian curvature equal to 1, which are also conformal to the Euclidean unit disk with the same conformal factor on the boundary, must be at least 4p. In otherwords, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We refer to this lower bound of total area as the Sphere Covering Inequality. The inequality and its 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 prove a conjecture proposed by Chang and Yang (Acta Math 159(34): 215-259, 1987) in the study of Nirenberg problem in conformal geometry.