Mean Curvature Flow of Spacelike Graphs

Details Mean Curvature Flow of Spacelike Graphs Mean Curvature Flow of Spacelike Graphs

2008-04-04

arxiv.org/abs/0804.0783v5

Mathematics

Differential Geometry

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

Scientific paper

10.1007/s00209-010-0768-4

We prove the mean curvature flow of a spacelike graph in $(\Sigma_1\times \Sigma_2, g_1-g_2)$ of a map $f:\Sigma_1\to \Sigma_2$ from a closed Riemannian manifold $(\Sigma_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold $(\Sigma_2,g_2)$ 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 $f:\Sigma_1\to \Sigma_2$ is trivially homotopic provided $f^*g_2<\rho g_1$ where $\rho=\min_{\Sigma_1}K_1/\sup_{\Sigma_2}K_2^+\geq 0$, in case $K_1>0$, and $\rho=+\infty$ in case $K_2\leq 0$. This largely extends some known results for $K_i$ constant and $\Sigma_2$ compact, obtained using the Riemannian structure of $\Sigma_1\times \Sigma_2$, 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.

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