*Accepted Paper*

**Inserted:** 25 oct 2017

**Last Updated:** 20 mar 2019

**Journal:** J. Reine Angew. Math. (Crelle's Journal).

**Year:** 2017

**Abstract:**

We prove that half spaces are the only stable nonlocal $s$-minimal cones in $\mathbb R^3$, 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 can not 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.

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