Hyperbolic mean curvature flow: Evolution of plane curves

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

26 pages

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