  We study the curve shortening flow on Riemann surfaces with finitely many
conformal conical singularities. If the initial curve is passing through the
singular points, then the evolution is governed by a degenerate quasilinear
parabolic equation. In this case, we establish short time existence,
uniqueness, and regularity of the flow. We also show that the evolving curves
stay fixed at the singular points of the surface and obtain some collapsing and
convergence results.
