Stable $s$-minimal cones in $\mathbb{R}^2$ are flat for $s \sim 0$

TL;DR

This paper proves that for very small s, the only stable s-minimal cones in two dimensions are flat half-planes, contrasting with known nontrivial stable cones for classical and near-one s regimes.

Contribution

It establishes the flatness of stable s-minimal cones in R^2 for small s, filling a gap in understanding the geometric structure of nonlocal minimal surfaces.

Findings

Only half-planes are stable cones for small s in R^2.

Nontrivial cones are stable for classical perimeter and s close to 1.

The result highlights a transition in stability behavior as s approaches zero.

Abstract

For $s \in (0,1)$ small, we show that the only cones in $\mathbb{R}^2$ stationary for the $s$-perimeter and stable in $\mathbb{R}^2 \setminus \{0\}$ are half-planes. This is in direct contrast with the case of the classical perimeter or the regime $s$ close to $1$, where nontrivial cones as $\{xy>0\} \subset \mathbb{R}^2$ are stable for inner variations.