A new construction of the solution for the curve shortening flow for curves of finite total (absolute) curvature

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. It 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, is globally defined, is well behaved and simpler to analyze. A solution of the CSF is obtained in the limit. Many classical qualitative results about the flow can be recovered and new ones can be proved. The construction works for non-immersed curves (an example is given) and suggests a novel numerical procedure for the computation of the flow (several experiments are shown). Finally, the construction shows that the singularities of the CSF prior to extinction are solely due to diffusion and not to the nonlinear nature of the system.