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

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 $$S_tS_z=S_{zz}$$ S t S 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.