Abstract
This article aims to contribute to the understanding of the curvature flow of curves in a higher-dimensional space. Evolution of curves in $ \mathbb{R}^m $ by their curvature is compared to the motion of hypersurfaces with constrained normal velocity. The special case of shrinking hyperspheres is further analyzed both theoretically and numerically by means of a semi-discrete scheme with discretization based on osculating circles. Computational examples of evolving spherical curves are provided along with the measurement of the experimental order of convergence.
Highlights
In physics, hypersurfaces and curves, describe interfaces between different phases, defects in crystalline structure of materials or boundaries of thin layers
We study the curvature flow of closed curves in Rm by means of the parametric approach
We introduce the notion of moving hypersurface as follows
Summary
Hypersurfaces and curves (mainly in R3), describe interfaces between different phases, defects in crystalline structure of materials or boundaries of thin layers (see [23, 19]). Their motion by curvature in various sense has been thoroughly studied, namely in two-dimensional case, where important theoretical results were obtained in [11, 10]. Primary: 53C44, 14Q05, 14J70; Secondary: 34K12, 35B51. Curvature flow, moving hypersurfaces, comparison principle, spherical curves, signed distance function
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