The fundamental solutions of the curve shortening problem via the Schwarz function

2021/03/10 by N. Robb McDonald, McDonald, Robb
Engineering · Mathematics · #30E99 #Advanced Numerical Analysis Techniques #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2103.06069

openalex publication_date 2021/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Curve shortening in the z-plane in which, at a given point on the curve, the normal velocity of the curve is equal to the curvature, is shown to satisfy S_tS_z=S_zz, where S(z,t) is the Schwarz function of the curve. This equation is shown to have a parametric solution from which the known explicit solutions for curve shortening flow; the circle, grim reaper, paperclip and hairclip, can be recovered.

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