Mean Curvature Flow of Spacelike Graphs

Abstract

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 $Ricc{i}_1>0$ to a complete Riemannian manifold $({\Sigma }_2,g_2)$ 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 $f:{\Sigma }_1\to {\Sigma }_2$ is trivially homotopic provided $f^{\ast }{g}_2<\rho g_1$ where $\rho ={\min }_{{\Sigma }_1}{K}_1/{\sup }_{{\Sigma }_2}{K}_2^{+}\ge 0$, in case $K_1>0$, and $\rho =+\infty$ in case $K_2\le 0$. 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.

version 5: Math.Z (online first 30 July 2010). version 4: 30 pages: we replace the condition $K_1\ge 0$ by the the weaker one $Ricc{i}_1\ge 0$. The proofs are essentially the same. We change the title to a shorter one. We add an application

References in corpus (4)

• Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces
• Long-time Existence and Convergence of Graphic Mean Curvature Flow in Arbitrary Codimension
• Singularity of Mean Curvature Flow of Lagrangian Submanifolds
• Graphic Bernstein Results in Curved Pseudo-Riemannian Manifolds

Cited by in corpus (3)

• Spacelike mean curvature flow
• Motivic double zeta values of odd weight
• A capillary problem for spacelike mean curvature flow in a cone of Minkowski space