  In this paper, we study the curve shortening flow (CSF) on Riemann surfaces.
We generalize Huisken's comparison function to Riemann surfaces and surfaces
with conic singularities. We reprove the Gage-Hamilton-Grayson theorem on
surfaces. We also prove that for embedded simple closed curves, CSF can not
touch conic singularities with cone angles $\leq \pi$.
