Curvature Motion in a Minkowski Plane

TL;DR

This paper investigates the curvature flow of convex curves in a Minkowski plane, demonstrating that such curves evolve towards a Minkowski circle while maintaining convexity and smoothness, similar to Euclidean cases.

Contribution

It extends known Euclidean curvature flow properties to Minkowski planes with smooth unit balls, showing convergence to Minkowski circles and preservation of convexity.

Findings

Convex curves remain convex until area vanishes.

Isoperimetric ratios tend to the minimal Minkowski value.

Curves converge to Minkowski circles as area approaches zero.

Abstract

In this paper we study the curvature flow of a curve in a plane endowed with a minkowskian norm whose unit ball is smooth. We show that many of the properties known in the euclidean case can be extended (with due adaptations) to this new situation. In particular, we show that simple, closed, strictly convex, smooth curves remain so until the area enclosed by them vanishes. Moreover, their isoperimetric ratios converge to the minimum possible value, only attained by the minkowskian circle - so these curves converge to a minkowskian "circular point" as the enclosed area approaches zero.