The sphere covering inequality and its applications
In this paper, we show that 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 \(4 \pi \). 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 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(3–4):215–259, 1987) in the study of Nirenberg problem in conformal geometry.
The authors would like to thank Professors Danielle Bartolucci, Alice Chang, Nassif Ghoussoub, Aleks Jevnikar, Yanyan Li, Fernando Marques, Richard Schoen, Paul Yang, etc for their interests and helpful comments on the earlier draft of the paper. The authors would also like to thank the anonymous referees for thoroughly checking the details of the manuscript and making valuble suggestions on the revision. The first author is partially supported by NSF Grant DMS-1601885. The second author is supported in part by NSF Grant DMS-1715850.
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