The sphere covering inequality and its applications
The sphere covering inequality and its applications
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DOI：
10.1007/s00222-018-0820-2
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发表时间：
2016-05
影响因子：
3.1
通讯作者：
C. Gui;Amir Moradifam
中科院分区：
文献类型：
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作者：
C. Gui;Amir Moradifam
本文证明了高斯曲率等于1的两个与欧几里得单位圆盘在边界上具有相同共形因子的不同曲面的总面积至少是。换句话说，经过适当的重新排列后，这些曲面的面积必须覆盖整个单位球面。我们将总面积的这个下界称为球面覆盖不等式。将该不等式及其推广应用于一些与Moser-Trudinger型不等式、平均场方程和Onsager涡旋等有关的公开问题，得到了最优结果。特别地，我们证明了张和杨(Acta Math 159(3-4)：215-259,1987)在研究保角几何中的Nirenberg问题时提出的一个猜想。
In this paper, we show that the total area of twodistinctsurfaces 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. 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.