  We define a new version of modified mean curvature flow (MMCF) in hyperbolic
space $\mathbb{H}^{n+1}$, which interestingly turns out to be the natural
negative $L^2$-gradient flow of the energy functional defined by De Silva and
Spruck in \cite{DS09}. We show the existence, uniqueness and convergence of the
MMCF of complete embedded star-shaped hypersurfaces with fixed prescribed
asymptotic boundary at infinity. As an application, we recover the existence
and uniqueness of smooth complete hypersurfaces of constant mean curvature in
hyperbolic space with prescribed asymptotic boundary at infinity, which was
first shown by Guan and Spruck.
