The Dirichlet problem for curve shortening flow
  We investigate the evolution of open curves with fixed endpoints under the curve shortening flow, which evolves curves in proportion to their curvature. Using a distance comparison of Huisken, we determine the long-term behavior of open curves with fixed endpoints evolving in certain convex domains on surfaces of constant curvature. Specifically, we show that such curves do not develop singularities, and evolve to geodesics. 
Paul T. Allen, Adam Layne, Katharine Tsukahara
math.DG math.AP