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

TL;DR

This paper classifies closed p-elastic curves on the sphere, showing existence only for specific p-values and demonstrating their instability for p in (0,1).

Contribution

It establishes the existence and instability of non-circular closed p-elastic curves on the sphere for p in (0,1), extending classical elastic curve theory.

Findings

Closed p-elastic curves exist only for p=2 or p in (0,1).

For p in (0,1), specific winding and closure conditions are characterized.

All closed p-elastic curves for p in (0,1) are proven to be unstable.

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.