  Let f:\Sigma_1 --> \Sigma_2 be a map between compact Riemannian manifolds of
constant curvature. This article considers the evolution of the graph of f in
the product of \Sigma_1 and \Sigma_2 by the mean curvature flow. Under suitable
conditions on the curvature of \Sigma_1 and \Sigma_2 and the differential of
the initial map, we show that the flow exists smoothly for all time. At each
instant t, the flow remains the graph of a map f_t and f_t converges to a
constant map as t approaches infinity. This also provides a regularity estimate
for Lipschtz initial data.
