Instability of Closed $p$-Elastic Curves in ${\mathbb{S}}^2$

Abstract

For $p\in \mathbb{R}$, we show that non-circular closed $p$-elastic curves in ${\mathbb{S}}^2$ exist only when $p=2$, in which case they are classical elastic curves, or when $p\in (0,1)$. In the latter case, we prove that for every pair of relatively prime natural numbers $n$ and $m$ satisfying $m<2n<\sqrt{2}m$, there exists a closed spherical $p$-elastic curve with non-constant curvature which winds around a pole $n$ times and closes up in $m$ periods of its curvature. Further, we show that all closed spherical $p$-elastic curves for $p\in (0,1)$ are unstable as critical points of the $p$-elastic energy.