We revisit the well-known Curve Shortening Flow for immersed curves in the $d$-dimensional Euclidean space. We exploit a fundamental structure of the problem to derive a new global construction of a solution, that is, a construction that is valid for all times and is insensitive to singularities. The construction is characterized by discretization in time and the approximant, while still exhibiting the possibile formation of finitely many singularities at a finite set of singular times, exists globally and is well behaved and simpler to analyze. A solution of the CSF is obtained in the limit. Estimates for a natural (geometric) norm involving length and total absolute curvature allow passage to the limit. Many classical qualitative results about the flow can be recovered by exploiting the simplicity of the approximant and new ones can be proved. The construction also suggests a numerical procedure for the computation of the flow which proves very effective as demonstrated by a series of numerical experiments scattered throughout the paper.

翻译：我们重新研究了 $d$ 维欧几里得空间中浸入曲线的著名曲线缩短流。利用该问题的一个基本结构，我们推导出一种新的全局解构造方法，该方法对所有时间均有效，并且对奇异性不敏感。该构造以时间离散化为特征，其近似函数虽然可能在有限个奇异时间点上形成有限个奇点，但全局存在、性态良好且更易于分析。通过取极限，可获得曲线缩短流的一个解。涉及长度和总绝对曲率的自然（几何）范数的估计，使得取极限成为可能。利用近似函数的简洁性，可以恢复许多关于该流的经典定性结果，并证明新的结论。该构造还提出了一种计算该流的数值方法，正如论文中散布的一系列数值实验所证明的那样，该方法非常有效。