Mean Curvature Flow of Spacelike Graphs

We prove the mean curvature flow of a spacelike graph in $(Σ_1\times Σ_2, g_1-g_2)$ of a map $f:Σ_1\to Σ_2$ from a closed Riemannian manifold $(Σ_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold $(Σ_2,g_2)$ 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 $f:Σ_1\to Σ_2$ is trivially homotopic provided $f^*g_2<ρg_1$ where $ρ=\min_{Σ_1}K_1/\sup_{Σ_2}K_2^+\geq 0$, in case $K_1>0$, and $ρ=+\infty$ in case $K_2\leq 0$. This largely extends some known results for $K_i$ constant and $Σ_2$ compact, obtained using the Riemannian structure of $Σ_1\times Σ_2$, 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.

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