论文标题
sobolev指标在多种有价值曲线的空间上
Sobolev metrics on spaces of manifold valued curves
论文作者
论文摘要
我们在多种有价值的开放和封闭的沉浸式曲线的空间上研究了订单$ n \ ge 2 $不变的Sobolev sobolev指标的完整性属性。特别是，对于几个重要的指标，我们表明Sobolev浸入量是测量和地理上完成的（因此，地球方程在全球范围内都有良好的序列）。以前，这些结果仅用于具有欧几里得空间中值的封闭曲线。对于恒定系数SOBOLEV指标上的类别，我们表明它们是指标不完整的，并且这种不完整仅来自完全消失的曲线（与低阶指标发生的“局部”故障不同）。
We study completeness properties of reparametrization invariant Sobolev metrics of order $n\ge 2$ on the space of manifold valued open and closed immersed curves. In particular, for several important cases of metrics, we show that Sobolev immersions are metrically and geodesically complete (thus the geodesic equation is globally well-posed). These results were previously known only for closed curves with values in Euclidean space. For the class of constant coefficient Sobolev metrics on open curves, we show that they are metrically incomplete, and that this incompleteness only arises from curves that vanish completely (unlike "local" failures that occur in lower order metrics).
