论文标题
$ \ mathbb {s}^2 $的关闭$ p $ - 弹性曲线的不稳定
Instability of Closed $p$-Elastic Curves in $\mathbb{S}^2$
论文作者
论文摘要
对于$ p \ in \ mathbb {r} $，我们表明仅在$ \ mathbb {s}^2 $中以$ p = 2 $存在$ p $ P $ - 弹性曲线，在这种情况下，它们是经典的弹性曲线，或者是$ p \ in（0,1,1）$。在后一种情况下，我们证明，对于每对相对主要的自然数量$ n $ $ n $和$ m $，满足$ m <2n <\ sqrt {2} \，m $，存在封闭的球形$ p $弹性曲线，具有非恒定率的曲率，在极点$ n $ n $ n $ n $ n $ m $ m $ $ m $ curvation中缠绕。此外，我们表明所有封闭的球形$ p $弹性曲线（0,1）$是$ p $弹性能的关键点不稳定。
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.
