Numerical approximation of curve evolutions in Riemannian manifolds

TL;DR

This paper develops stable numerical schemes for approximating curve evolutions such as curvature flow, diffusion, and elastic flow in conformally flat Riemannian manifolds, including hyperbolic and elliptic geometries, with applications to axisymmetric hypersurfaces.

Contribution

It introduces variational approximation schemes for curve evolutions in conformally flat Riemannian manifolds, ensuring stability and good mesh point distribution.

Findings

Schemes successfully applied to hyperbolic and elliptic geometries.

Numerical experiments demonstrate scheme stability and mesh quality.

Applicable to geometric evolution equations for axisymmetric hypersurfaces.

Abstract

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.