Numerical approximation of curve evolutions in Riemannian manifolds

We introduce variational approximations for curve evolutions in two-dimensional Riemannian manifolds that are conformally flat, i.e. conformally equivalent to the Euclidean plane. Examples include the hyperbolic plane, the hyperbolic disc and the elliptic plane, as well as any conformal parameterization of a two-dimensional manifold in Rd⁠, d⩾3⁠. In these spaces we introduce stable numerical schemes for curvature flow and curve diffusion, and we also formulate schemes for elastic flow. Variants of the schemes can also be applied to geometric evolution equations for axisymmetric hypersurfaces in Rd⁠. 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.