We prove that half spaces are the only stable nonlocal s-minimal cones in ${ℝ}^3$, for $s\in (0,1)$ sufficiently close to 1. This is the first classification result of stable s-minimal cones in dimension higher than two. Its proof cannot rely on a compactness argument perturbing from $s=1$. In fact, our proof gives a quantifiable value for the required closeness of s to 1. We use the geometric formula for the second variation of the fractional s-perimeter, which involves a squared nonlocal second fundamental form, as well as the recent BV estimates for stable nonlocal minimal sets.

中文翻译：

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ℝ3中稳定的s最小锥面对于s〜1是平坦的

我们证明半空间是唯一稳定的非局部s-最小锥${ℝ}^3$，对于 $s\in （0，\mathrm{1个}）$充分接近于1。这是尺寸大于2的稳定s-最小锥的第一个分类结果。它的证明不能依赖于从$s=\mathrm{1个}$。实际上，我们的证明给出了s与1的要求接近度的可量化值。我们使用分数s- perimeter的第二个变化的几何公式，其中包括平方非局部第二基本形式以及最近的BV估计用于稳定的非局部极小集。