An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane $\mathbb{R}^{2}_{1}$

TL;DR

This paper studies the evolution of spacelike graphic curves in the Lorentz-Minkowski plane under an anisotropic inverse mean curvature flow, proving long-term existence and convergence to a hyperbola after rescaling.

Contribution

It introduces an anisotropic inverse mean curvature flow for spacelike curves in Lorentz-Minkowski space and proves their global existence and asymptotic convergence.

Findings

Flow exists for all time

Curves converge to a hyperbola after rescaling

Convergence is smooth and characterized by a constant function

Abstract

In this paper, we consider the evolution of spacelike graphic curves defined over a piece of hyperbola $\mathscr{H}^{1}(1)$, of center at origin and radius $1$, in the $2$ dimensional Lorentz-Minkowski plane $\mathbb{R}^{2}_{1}$ along an anisotropic inverse mean curvature flow with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic curves converge smoothly to a piece of hyperbola of center at origin and prescribed radius, which actually corresponds to a constant function defined over the piece of $\mathscr{H}^{1}(1)$, as time tends to infinity.