The Curve Shortening Flow in the Metric-Affine Plane

Vladimir Rovenski

doi:10.3390/math8050701

Abstract

We investigated, for the first time, the curve shortening flow in the metric-affine plane and prove that under simple geometric condition (when the curvature of initial curve dominates the torsion term) it shrinks a closed convex curve to a “round point” in finite time. This generalizes the classical result by M. Gage and R.S. Hamilton about convex curves in a Euclidean plane.

Highlights

The one-dimensional mean curvature flow is called the curve shortening flow (CSF), because it is the negative L2 -gradient flow of the length of the interface, and it is used in modeling the dynamics of melting solids
The main contribution of this paper is a geometrical proof of convergence of the flow (3) for convex closed curves
The paper carries out the initial step of the investigation of interest, when the curvature of initial curve dominates the torsion term so that the behavior is essentially the same as for the classical CSF

Summary

Introduction

The one-dimensional mean curvature flow is called the curve shortening flow (CSF), because it is the negative L2 -gradient flow of the length of the interface, and it is used in modeling the dynamics of melting solids. (b) Rescaling in order to keep the length constant, the flow converges exponentially fast to a circle in C ∞ This theorem and further result by M.A. Grayson, [4] (that the flow moves any closed embedded in the Euclidean plane curve in a finite time to a convex curve) have many generalizations and applications in natural and computer sciences. The approach of [1] to the normalized flow of (2) in the contracting case still works without the positivity of Ψ, see ([1] Remark 3.14) Based on this result and Theorem 2, we obtain the following result, generalizing Theorem 1(b). Theorem 2 can be extended to the case of non-constant contorsion tensor T of small norm, but we can not reject the Condition 1 for Theorem 3, since its proof is based on the result for the normalized ACEF, see [1], where Ψ depends only on θ

Proof of Theorem 2

Proof of Theorem 3

Conclusion