We study the curvature flow of planar nonconvex lens-shaped domains, considered as special symmetric networks with two triple junctions. We show that the evolving domain becomes convex in finite time; then it shrinks homothetically to a point, as proved in [Schnürer, Azouani, Georgi, Hell, Jangle, Köller, Marxen, Ritthaler, Sáez, Schulze and Smith, Trans. Amer. Math. Soc.]. Our theorem is the analog of the result of Grayson [J. Diff. Geom. 26: 285–314, 1987] for curvature flow of closed planar embedded curves.
© Walter de Gruyter Berlin · New York 2011