Hyperbolic mean curvature flow: Evolution of plane curves

Kong, De-Xing; Liu, Kefeng; Wang, Zeng-Gui

Hyperbolic mean curvature flow: Evolution of plane curves

Authors

Kong, De-Xing

Liu, Kefeng

Wang, Zeng-Gui

Description

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 $\mathbb{R}^{1,1}$.
26 pages