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Mean Curvature Flow of Closed Curves Evolving in Two Dimensional Manifolds

Miroslav Kolar, Daniel Sevcovic

TL;DR

The paper studies mean-curvature-type evolution of closed curves constrained to evolve on two-dimensional manifolds embedded or immersed in three-dimensional space, formulating a nonlinear parabolic system for the curve position and proving local well-posedness via analytic semigroups.A direct Lagrangian parametrization is used, with velocities decomposed into normal, binormal, and tangential components and a constraint force used to keep curves on the target surface, yielding a length-decreasing flow on the ambient surface.Numerical approximation is developed through a flowing finite-volume method with asymptotically uniform tangential redistribution, and demonstrated on several illustrative geometries (torus, Klein bottle-like immersion, genus-0 surfaces with humps), highlighting stability and convergence properties.The results provide a rigorous foundation for simulating curvature-driven curve flows on complex surfaces with potential applications to materials science problems such as dislocation dynamics and nanofiber formation.

Abstract

We investigate the motion of a family of closed curves evolving according to the geometric evolution law on a given two dimensional manifold which is embedded or immersed in the three-dimensional Euclidean space. We derive a system of nonlinear parabolic equations describing the motion of curves belonging to a given two-dimensional manifold. Using the abstract theory of analytic semiflows, we prove the local existence, uniqueness of Hölder smooth solutions to the governing system of nonlinear parabolic equations for the position vector parametrization of evolving curves. We apply the method of flowing finite volumes in combination with the methods of lines for numerical approximation of the governing equations. Qualitative analytical results are illustrated by various numerical experiments.

Mean Curvature Flow of Closed Curves Evolving in Two Dimensional Manifolds

TL;DR

The paper studies mean-curvature-type evolution of closed curves constrained to evolve on two-dimensional manifolds embedded or immersed in three-dimensional space, formulating a nonlinear parabolic system for the curve position and proving local well-posedness via analytic semigroups.A direct Lagrangian parametrization is used, with velocities decomposed into normal, binormal, and tangential components and a constraint force used to keep curves on the target surface, yielding a length-decreasing flow on the ambient surface.Numerical approximation is developed through a flowing finite-volume method with asymptotically uniform tangential redistribution, and demonstrated on several illustrative geometries (torus, Klein bottle-like immersion, genus-0 surfaces with humps), highlighting stability and convergence properties.The results provide a rigorous foundation for simulating curvature-driven curve flows on complex surfaces with potential applications to materials science problems such as dislocation dynamics and nanofiber formation.

Abstract

We investigate the motion of a family of closed curves evolving according to the geometric evolution law on a given two dimensional manifold which is embedded or immersed in the three-dimensional Euclidean space. We derive a system of nonlinear parabolic equations describing the motion of curves belonging to a given two-dimensional manifold. Using the abstract theory of analytic semiflows, we prove the local existence, uniqueness of Hölder smooth solutions to the governing system of nonlinear parabolic equations for the position vector parametrization of evolving curves. We apply the method of flowing finite volumes in combination with the methods of lines for numerical approximation of the governing equations. Qualitative analytical results are illustrated by various numerical experiments.
Paper Structure (12 sections, 5 theorems, 48 equations, 7 figures)

This paper contains 12 sections, 5 theorems, 48 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Lagrangian description of evolving curves
  3. Evolution of closed curves on a closed surface without boundary
  4. Evolution of curves on embedded manifolds
  5. Evolution of curves on immersed manifolds
  6. Existence and uniqueness of classical Hölder smooth solutions
  7. Flowing finite volumes numerical discretization scheme
  8. Examples
  9. Evolution of knotted curves on torus
  10. Attraction of curves by a torus surface
  11. Evolution of simple curves on surface with genus 0 with humps
  12. Conclusion

Key Result

Theorem 1

Suppose that the external force ${\mathbf F}$ is given by ${\mathbf T}=\partial_s{\mathbf X}$. Here $h:\mathbb{R}\to\mathbb{R}$ is a smooth function, $h(0)=0$. If the initial curve $\Gamma_0 \subset {\mathcal{M}}$ then the family of curves $\Gamma_t$, for $t>0$, evolving with respect to the geometric equation (eq:ab) belongs to the manifold ${\mathcal{M}}=

Figures (7)

  • Figure 1: A knotted curve a) belonging to the embedded torus surface b).
  • Figure 2: a) An initial knotted curve belonging to the immersed Klein bottle surface b)
  • Figure 3: Discretization of a segment of a 3D curve on a surface by the method of flowing finite volumes.
  • Figure 4: Time evolution of a knotted Fourier curve belonging to the orientable torus surface with parameters $r=1, R=4$, and $k=2$, $l=3$ (see (\ref{['parameterization']})).
  • Figure 5: Time evolution of initially inflated knotted curve belonging to the orientable torus surface with parameters $r=1, R=4$. Topologically more complex case corresponding to the choice $k=3$, $l=5$ in initial condition (\ref{['parameterization']}).
  • ...and 2 more figures

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Theorem 3
  • proof
  • ...and 3 more