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

Xavier Cabre; Eleonora Cinti; Joaquim Serra

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arXiv · 1710.08722

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

Xavier Cabre·Eleonora Cinti·Joaquim Serra

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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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Xavier Cabre, Eleonora Cinti, Joaquim Serra. 2017-10-24. Stable $s$-minimal cones in $\mathbb{R}^3$ are flat for $s\sim 1$. https://arxiv.org/abs/1710.08722

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