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<p>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.</p>

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Mean curvature flow of closed curves evolving in two dimensional manifolds

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年2月11日

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  • Miroslav Kolář,
  • Daniel Ševčovič

摘要

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.