  Motivated by previous study on mean curvature flow and prescribed mean
curvature flow on spatially compact space or asymptotically flat spacetime, in
this work we will find sufficient conditions for the short time existence of
prescribed mean curvature flow on a Lorentz manifold with a smooth time
function starting from a complete noncompact spacelike hypersurface. Long time
existence and convergence will also be discussed. Results will be applied to
study some prescribed mean curvature flows inside the future of the origin in
the Minkowski spacetime. Examples of spacetime related to the existence and
convergence results near the future null infinity of the Schwarzschild
spacetime are also discussed.
