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arXiv 2108.08686math.DG
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Lorentz-Minkowski空间$\mathbb{R}^{n+1}_{1}$时间锥中的逆高斯曲率流

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

Ya Gao, Jing Mao

备注：
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
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摘要：

本文研究了定义在双曲平面$\mathscr{H}^{n}(1)$的凸片上的类空图超曲面在$(n+1)$维Lorentz-Minkowski空间$\mathbb{R}^{n+1}_{1}$中沿逆Gauss曲率流的演化（即，演化速度等于高斯曲率的$(-1/n)$次幂），并证明了该流始终存在.此外，我们可以证明，经过适当的重新标度，不断发展的类空图形超曲面光滑收敛到一片定义在一块$\mathscr{H}^{n}(1)$的正常数函数的类空图的时间趋于无穷大。

英文摘要：

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.

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