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

TL;DR

This paper derives fundamental solutions for the curve shortening problem using the Schwarz function, providing a unified framework that recovers known explicit solutions like the circle and grim reaper.

Contribution

It introduces a novel PDE involving the Schwarz function that characterizes curve shortening flow and derives explicit solutions from it.

Findings

Unified PDE framework for curve shortening solutions

Recovery of classical solutions such as circle and grim reaper

New parametric solutions derived from the PDE

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.