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

created by cinti on 25 Oct 2017
modified on 06 Feb 2018

[BibTeX]

Preprint

Inserted: 25 oct 2017
Last Updated: 6 feb 2018

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.