Stable $s$-minimal cones in $\mathbb{R}^2$ are flat for $s \sim 0$
Michele Caselli
TL;DR
The paper proves that for $s$ sufficiently small, the only $s$-minimal cones in $\mathbb{R}^2$ that are stable in $\mathbb{R}^2\setminus\{0\}$ are half-planes, highlighting a purely nonlocal phenomenon contrasting with the classical ($s\approx1$) case where nontrivial cones can be stable. The authors develop a BV-estimate for stable $s$-minimal surfaces as $s\to0$, combine it with a Hardy-inequality-based saturation argument, and show that any cone must have a single sector, hence be flat. They extend the result to cones of finite Morse index outside the origin via a weighted second-variation analysis, implying the same flatness conclusion under that broader condition. An appendix confirms the instability of the cross cone $X=\{xy>0\}$ for small $s$ in $\mathbb{R}^2\setminus\{0\}$, underscoring the nonlocal nature of the phenomenon. Overall, the work delineates a sharp, low-$s$ regime where stability enforces flatness in two dimensions, with implications for fractional widths and regularity of $s$-minimal surfaces.
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.