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

Numerical approximation of curve evolutions in Riemannian manifolds