A curvature flow that deforms curves to an embedded target

In this paper we introduce the target flow —a specific curve shortening flow withan ambient forcing term—that, given an embedded (not necessarily convex) target curveη : S → R2, will attempt to evolve a given source curve to that target. The motivationfor this flow is to address a question of Yau. Our main result is that the target flow withuniformly normal graphical data converges smoothly to the target, broadening the class ofknown sources and targets such that Yau’s problem has a solution.