Abstract. The curve shortening problem for a graph under Robin boundary condition is studied in this paper. The large time behavior of the global solution is shown to depend critically on the parameters in the boundary condition. Some asymptotic behavior of the solution is also discussed. Mathematics Subject Classification (2010

A saddle-point property of the self-similar solutions in the curve shortening flow was conjectured by Abresch and Langer and confirmed by Au. An improvement on Au's solution is presented.