The asymptotic Plateau problem asks for the existence of smooth complete hypersurfaces of constant mean curvature with prescribed asymptotic boundary at infinity in the hyperbolic space $\mathbb{H}^{n+1}$. The modified mean curvature flow (MMCF) was firstly introduced by Xiao and the second author a few years back, and it provides a tool using geometric flow to find such hypersurfaces with constant mean curvature in $\mathbb{H}^{n+1}$. Similar to the usual mean curvature flow, the MMCF is the natural negative $L^2$-gradient flow of the area-volume functional $\mathcal{I}(\Sigma)=A(\Sigma)+\sigma V(\Sigma)$ associated to a hypersurface $\Sigma$. In this paper, we prove that the MMCF starting from an entire locally Lipschitz continuous radial graph exists and stays radially graphic for all time. In general one cannot expect the convergence of the flow as it can be seen from the flow starting from a horosphere (whose asymptotic boundary is degenerate to a point).

中文翻译：

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双曲空间中整个局部 Lipschitz 径向图的修正平均曲率流

渐近高原问题要求在双曲空间 $\mathbb{H}^{n+1}$ 中存在具有指定渐近边界的恒定平均曲率的光滑完全超曲面。修正平均曲率流（MMCF）是几年前由肖和第二作者首先提出的，它提供了一种使用几何流在$\mathbb{H}^{n+1 中找到具有恒定平均曲率的超曲面的工具}$。类似于通常的平均曲率流，MMCF 是面积-体积函数 $\mathcal{I}(\Sigma)=A(\Sigma)+\sigma V(\ Sigma)$ 与超曲面 $\Sigma$ 相关联。在本文中，我们证明了从整个局部 Lipschitz 连续径向图开始的 MMCF 存在并始终保持径向图形。