Grim raindrop: A translating solution to curve diffusion flow

We show the existence of a properly immersed translating solution to curve diffusion flow in the plane. Curve diffusion flow is a higher order version of curve shortening flow, namely ( d X d t ) ⊥ = − κ s s N . {equation*} ( dX/dt ) ⊥=-κ ₛₛN. {equation*}