Translating solutions for a class of quasilinear parabolic initial boundary value problems in Lorentz-Minkowski plane $\mathbb{R}^{2}_{1}$

TL;DR

This paper studies the evolution of spacelike curves in Lorentz-Minkowski plane under geometric flows, proving long-term existence and convergence to specific curves, extending understanding of quasilinear parabolic problems in Lorentzian geometry.

Contribution

It introduces a framework for analyzing geometric flows of spacelike curves in Lorentz-Minkowski space, proving existence and convergence results for these flows.

Findings

Flow exists for all time

Curves converge to spacelike lines or Grim Reaper curves

Extends geometric flow analysis to Lorentzian settings

Abstract

In this paper, we investigate the evolution of spacelike curves in Lorentz-Minkowski plane $\mathbb{R}^{2}_{1}$ along prescribed geometric flows (including the classical curve shortening flow or mean curvature flow as a special case), which correspond to a class of quasilinear parabolic initial boundary value problems, and can prove that this flow exists for all time. Moreover, we can also show that the evolving spacelike curves converge to a spacelike straight line or a spacelike Grim Reaper curve as time tends to infinity.