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

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