We prove the mean curvature flow of a spacelike graph in $(��_1\times ��_2, g_1-g_2)$ of a map $f:��_1\to ��_2$ from a closed Riemannian manifold $(��_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold $(��_2,g_2)$ with bounded curvature tensor and derivatives, and with sectional curvatures satisfying $K_2\leq K_1$, remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumption $K_2\leq K_1$, that if $K_1>0$, or if $Ricci_1>0$ and $K_2\leq -c$, $c>0$ constant, any map $f:��_1\to ��_2$ is trivially homotopic provided $f^*g_20$, and $��=+\infty$ in case $K_2\leq 0$. This largely extends some known results for $K_i$ constant and $��_2$ compact, obtained using the Riemannian structure of $��_1\times ��_2$, and also shows how regularity theory on the mean curvature flow is simpler and more natural in pseudo-Riemannian setting then in the Riemannian one.

version 5: Math.Z (online first 30 July 2010). version 4: 30 pages: we replace the condition $K_1\geq 0$ by the the weaker one $Ricci_1\geq 0$. The proofs are essentially the same. We change the title to a shorter one. We add an application