UC Santa Cruz

In \cite{GS00}, B. Guan and J. Spruck showed the existence of smooth radial graphs of constant mean curvature with prescribed $C^0$, star-shaped boundary at infinity using elliptic PDE methods and the maximum principle. Surfaces of constant mean curvature are critical points of an area functional with a volume constraint.

In \cite{DS09}, D. De Silva and J. Spruck showed the same result mentioned above in \cite{GS00} using variational methods. It is a natural question then to ask whether we can approach this problem using the negative gradient flow of that area-volume functional. Such a flow, called \textbf{modified mean curvature flow}, was first introduced by L. Lin and L. Xiao in \cite{LX}. There they showed, starting with a star-shaped hypersurface with a global $C^1$ bound, the longtime existence of the modified mean curvature flow. Moreover, they recovered the previous results by showing the flow converges to a stationary solution.

This work is inspired by these three works. Here, we show the longtime existence of a smooth modified mean curvature flow of hypersurfaces in hyperbolic space if the initial hypersurface is locally Lipschitz and star-shaped. This result can be considered as a generalization of the main theorem of \cite{U03} by P. Unterberger, in which they show a longtime existence result of \textbf{mean curvature flow} in the same ambient and initial setting. It's also a hyperbolic version of the nonparametric mean curvature flow in Euclidean space studied by K. Ecker and G. Huisken in \cite{EH91}. There they found a locally Lipschitz vertical graph moving by its mean curvature becomes a smooth vertical graph for all time.\