Stable s-minimal cones in ℝ3 are flat for s ~ 1

Abstract

Abstract We prove that half spaces are the only stable nonlocal s-minimal cones in ℝ 3 {\mathbb{R}^{3}} , for s ∈ ( 0 , 1 ) {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 {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.