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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 firstclassification result of stable $s$-minimal cones in dimension higher than two.Its proof can not rely on a compactness argument perturbing from $s=1$. Infact, 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, aswell as the recent BV estimates for stable nonlocal minimal sets.

Stable $s$-minimal cones in $\mathbb{R}^3$ are flat for $s\sim 1$