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 than a solution of the CSF. A solution of the latter 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维欧氏空间中浸入曲线的经典曲线缩短流。通过利用该问题的基本结构，我们推导出一种新的全局解构造方法，该构造方法对所有时间均有效且对奇点不敏感。该构造以时间离散化为特征，其逼近函数虽可能在有限个奇异时刻形成有限个奇点，但全局存在、性质良好且比CSF解更易于分析，而CSF解可通过极限过程获得。包含长度与总绝对曲率的自然（几何）范数估计保证了极限过程的可行性。利用逼近函数的简洁性，可重现该流动的诸多经典定性结果并证明新结论。该构造还提出了数值计算流程的有效算法，通过文中散布的一系列数值实验证明了其高效性。