  We present variational approximations of boundary value problems for
curvature flow (curve shortening flow) and elastic flow (curve straightening
flow) in two-dimensional Riemannian manifolds that are conformally flat. For
the evolving open curves we propose natural boundary conditions that respect
the appropriate gradient flow structure. Based on suitable weak formulations we
introduce finite element approximations using piecewise linear elements. For
some of the schemes a stability result can be shown. The derived schemes can be
employed in very different contexts. For example, we apply the schemes to the
Angenent metric in order to numerically compute rotationally symmetric
self-shrinkers for the mean curvature flow. Furthermore, we utilise the schemes
to compute geodesics that are relevant for optimal interface profiles in
multi-component phase field models.
