  In this paper, we study the regular geometric behavior of the mean curvature
flow (MCF) of submanifolds in the standard Gaussian metric space $({\mathbb
R}^{m+p},e^{-|x|^2/m}\ol g)$ where $({\mathbb R}^{m+p},\ol g)$ is the standard
Euclidean space and $x\in{\mathbb R}^{m+p}$ denotes the position vector. Note
that, as a special Riemannian manifold, $({\mathbb R}^{m+p},e^{-|x|^2/m}\ol g)$
has an unbounded curvature. Up to a family of diffeomorphisms on $M^m$, the
mean curvature flow we considered here turns out to be equivalent to a special
variation of the ``{\em conformal mean curvature flow}\,'' which we have
introduced previously. The main theorem of this paper indicates, geometrically,
that any immersed compact submanifold in the standard Gaussian space, with the
square norm of the position vector being not equal to $m$, will blow up at a
finite time under the mean curvature flow, in the sense that either the
position or the curvature blows up to infinity; Moreover, by this main theorem,
the interval $[0,T)$ of time in which the flowing submanifolds keep regular has
some certain optimal upper bound, and it can reach the bound if and only if the
initial submanifold either shrinks to the origin or expands uniformly to
infinity under the flow. Besides the main theorem, we also obtain some other
interesting conclusions which not only play their key roles in proving the main
theorem but also characterize in part the geometric behavior of the flow, being
of independent significance.
