Inverse Gauss curvature flow in a time cone of Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$

In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$ along the inverse Gauss curvature flow (i.e., the evolving speed equals the $(-1/n)$-th power of the Gaussian curvature) 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 hypersurfaces converge smoothly to a piece of the spacelike graph of a positive constant function defined over the piece of $\mathscr{H}^{n}(1)$ as time tends to infinity.
Comment: This work has been announced in our previous work [arXiv:2104.10600v4]. 31 pages. Comments are welcome. arXiv admin note: text overlap with arXiv:2106.05973