Mean Curvature Flow of Spacelike Graphs

We prove the mean curvature flow of a spacelike graph in $(\Sigma_1\times \Sigma_2, g_1-g_2)$ofamap$f:\Sigma_1\to \Sigma_2$ from a closed Riemannian manifold $({\mathrm{\Sigma }}_1,g_1)$ with $Ricc{i}_1>0$ to a complete Riemannian manifold $({\mathrm{\Sigma }}_2,g_2)$ with bounded curvature tensor and derivatives, and with sectional curvatures satisfying $K_2\le K_1$, remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumption $K_2\le K_1$, that if $K_1>0$, or if $Ricc{i}_1>0$ and $K_2\le -c$, $c>0$ constant, any map $f:{\mathrm{\Sigma }}_1\to {\mathrm{\Sigma }}_2$ is trivially homotopic provided $f^{\ast }{g}_2<\rho g_1$ where $\rho ={\min }_{{\mathrm{\Sigma }}_1}{K}_1\mathrm{/}{\sup }_{{\mathrm{\Sigma }}_2}{K}_2^{+}\ge 0$, in case $K_1>0$, and $\rho =+\mathrm{\infty }$ in case $K_2\le 0$. This largely extends some known results for $K_i$ constant and ${\mathrm{\Sigma }}_2$ compact, obtained using the Riemannian structure of ${\mathrm{\Sigma }}_1\times {\mathrm{\Sigma }}_2$, 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.