A note on traveling waves for area-preserving geometric flows

Abstract

The traveling waves for area-preserving geometric flows will be discussed. For an evolving plane curve, which is governed by the area-preserving curvature flow and has two endpoints moving freely on the $x$-axis with generating constant contact angles, we prove that there exists only a convex traveling wave. In the proof, a boundary value problem for an angle function associated with the Gauss map of the profile curve is analyzed by means of a shooting argument.