We prove the mean curvature flow of a spacelike graph in $(Sigma_1times Sigma_2, g_1-g_2)$ of a map $f:Sigma_1to Sigma_2$ from a closed Riemannian manifold $(Sigma_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold $(Sigma_2,g_2)$ with bound
ed curvature tensor and derivatives, and with sectional curvatures satisfying $K_2leq K_1$, remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumption $K_2leq K_1$, that if $K_1>0$, or if $Ricci_1>0$ and $K_2leq -c$, $c>0$ constant, any map $f:Sigma_1to Sigma_2$ 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 $K_2leq 0$. This largely extends some known results for $K_i$ constant and $Sigma_2$ compact, obtained using the Riemannian structure of $Sigma_1times Sigma_2$, 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.
We generalize a Bernstein-type result due to Albujer and Alias, for maximal surfaces in a curved Lorentzian product 3-manifold of the form $Sigma_1times mathbb{R}$, to higher dimension and codimension. We consider $M$ a complete spacelike graphic sub
manifold with parallel mean curvature, defined by a map $f: Sigma_1to Sigma_2$ between two Riemannian manifolds $(Sigma_1^m, g_1)$ and $(Sigma^n_2, g_2)$ of sectional curvatures $K_1$ and $K_2$, respectively. We take on $Sigma_1times Sigma_2$ the pseudo-Riemannian product metric $g_1-g_2$. Under the curvature conditions, $mathrm{Ricci}_1 geq 0$ and $K_1geq K_2$, we prove that, if the second fundamental form of $M$ satisfies an integrability condition, then $M$ is totally geodesic, and it is a slice if $mathrm{Ricci}_1(p)>0$ at some point. For bounded $K_1$, $K_2$ and hyperbolic angle $theta$, we conclude $M$ must be maximal. If $M$ is a maximal surface and $K_1geq K_2^+$, we show $M$ is totally geodesic with no need for further assumptions. Furthermore, $M$ is a slice if at some point $pin Sigma_1$, $K_1(p)> 0$, and if $Sigma_1$ is flat and $K_2<0$ at some point $f(p)$, then the image of $f$ lies on a geodesic of $Sigma_2$.
Given $(bar{M},Omega)$ a calibrated Riemannian manifold with a parallel calibration of rank $m$, and $M^m$ an immersed orientable submanifold with parallel mean curvature $H$ we prove that if $cos theta$ is bounded away from zero, where $theta$ is th
e $Omega$-angle of $M$, and if $M$ has zero Cheeger constant, then $M$ is minimal. In the particular case $M$ is complete with $Ricc^Mgeq 0$ we may replace the boundedness condition on $cos theta$ by $cos thetageq Cr^{-beta}$, when $rto +infty$, where $ 0leqbeta <1 $ and $C > 0$ are constants and $r$ is the distance function to a point in $M$. Our proof is surprisingly simple and extends to a very large class of submanifolds in calibrated manifolds, in a unified way, the problem started by Heinz and Chern of estimating the mean curvature of graphic hypersurfaces in Euclidean spaces. It is based on a estimation of $|H|$ in terms of $costheta$ and an isoperimetric inequality. We also give some conditions to conclude $M$ is totally geodesic. We study some particular cases.
This is a survey of our work on spacelike graphic submanifolds in pseudo-Riemannian products, namely on Heinz-Chern and Bernstein-Calabi results and on the mean curvature flow, with applications to the homotopy of maps between Riemannian manifolds.
