We introduce variational approximations for curve evolutions in
two-dimensional Riemannian manifolds that are conformally flat, i.e.\
conformally equivalent to the Euclidean space. Examples include the hyperbolic
plane, the hyperbolic disk, the elliptic plane as well as any conformal
parameterization of a two-dimensional surface in ${\mathbb R}^d$, $d\geq 3$. In
these spaces we introduce stable numerical schemes for curvature flow and curve
diffusion, and we also formulate a scheme for elastic flow. Variants of the
schemes can also be applied to geometric evolution equations for axisymmetric
hypersurfaces in ${\mathbb R}^d$. Some of the schemes have very good properties
with respect to the distribution of mesh points, which is demonstrated with the
help of several numerical computations.