Hyperbolic mean curvature flow: Evolution of plane curves

TL;DR

This paper studies the hyperbolic mean curvature flow of closed plane curves, showing finite-time convergence to a point and exploring its connection to relativistic string equations in Minkowski space.

Contribution

It introduces the hyperbolic mean curvature flow for plane curves, analyzes finite-time existence, and links it to relativistic string evolution equations.

Findings

Solutions exist only on finite time intervals.

Solutions converge to a point as time approaches the maximum.

Connection established between curvature flow and relativistic strings.

Abstract

In this paper we investigate the one-dimensional hyperbolic mean curvature flow for closed plane curves. We show that there exists a class of initial velocities such that the solution of the corresponding initial value problem exists only at a finite time interval $[0,T_{\max})$ and when $t$ goes to $T_{\max}$, the solution converges to a point. We also discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-time $\mathbb{R}^{1,1}$.