Riemannian流形中曲线演化的数值近似
Numerical approximation of curve evolutions in Riemannian manifolds
AI总结：
本文提出了一种在二维仿射平坦流形中曲线演化的变分近似方法，适用于双曲平面、双曲盘、椭圆平面及三维空间中二维表面的共形参数化。研究了曲率流和曲线扩散的稳定数值方案，并提出了弹性流的方案，同时该方案可扩展至三维空间中轴对称超曲面的几何演化方程。
摘要：
我们介绍了在二维仿射平坦流形中曲线演化的变分近似方法，即与欧几里得空间共形等价的流形，例如双曲平面、双曲盘、椭圆平面以及三维空间${\mathbb R}^d$中任意二维表面的共形参数化。在这些空间中，我们引入了曲率流和曲线扩散的稳定数值方案，并提出了弹性流的方案。这些方案的变种也可用于三维空间${\mathbb R}^d$中轴对称超曲面的几何演化方程。一些方案在网格点分布方面具有非常良好的性质，这通过几个数值计算得到了验证。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
