Mean Curvature Flow of Closed Curves Evolving in Two Dimensional Manifolds

TL;DR

This paper studies the evolution of closed curves on two-dimensional manifolds within three-dimensional space, deriving equations, proving local existence and uniqueness of solutions, and demonstrating numerical methods and experiments.

Contribution

It introduces a system of nonlinear parabolic equations for curve evolution on manifolds and proves local well-posedness using analytic semiflows.

Findings

Established local existence and uniqueness of smooth solutions.

Developed numerical methods combining flowing finite volumes and lines.

Validated results with numerical experiments.

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 H\"older 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.