论文标题
具有广义诺伊曼边界条件的平面曲线的高阶曲率流动
Higher order curvature flows of plane curves with generalised Neumann boundary conditions
论文作者
论文摘要
我们考虑抛物线多结的扩散和$ l^2 $ - $ m $ th arclength curvature of Curvature of Curvature of Curvature for Neumann边界条件的常规闭合曲线。在多谐调案例中，我们证明，如果初始曲线的曲率在$ l^2 $中很小，则不断发展的曲线在$ c^\ infty $ topology中成倍收敛到直线线段。只要初始曲线的能量足够小，$ l^2 $ - 级别的流量也显示出相同的行为。在每种情况下，较小的条件仅取决于$ m $。
We consider the parabolic polyharmonic diffusion and $L^2$-gradient flows of the $m$-th arclength derivative of curvature for regular closed curves evolving with generalised Neumann boundary conditions. In the polyharmonic case, we prove that if the curvature of the initial curve is small in $L^2$, then the evolving curve converges exponentially in the $C^\infty$ topology to a straight horizontal line segment. The same behaviour is shown for the $L^2$-gradient flow provided the energy of the initial curve is sufficiently small. In each case the smallness conditions depend only on $m$.
