This note treats several problems for the fractional perimeter or $ s $-perimeter on the sphere. The spherical fractional isoperimetric inequality is established. It turns out that the equality cases are exactly the spherical caps. Furthermore, the convergence of fractional perimeters to the surface area as $ s \nearrow 1 $ is proven. It is shown that their limit as $ s \searrow -\infty $ can be expressed in terms of the volume.

中文翻译：

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球体上的分数周长

本笔记处理球体上的分数周长或 $s $-perimeter 的几个问题。建立球面分数等周不等式。事实证明，等式情况正是球冠。此外，分数周长收敛到表面积为 $ s \nearrow 1 $ 被证明。结果表明，它们的极限为 $ s \searrow -\infty $ 可以用体积来表示。