Parabolic Omori-Yau maximum principle for mean curvature flow and some applications

We derive a parabolic version of Omori-Yau maximum principle for a proper mean curvature flow when the ambient space has lower bound on $\ell$-sectional curvature. We apply this to show that the image of Gauss map is preserved under a proper mean curvature flow in euclidean spaces with uniform bounded second fundamental forms. This generalizes the result of Wang \cite{Wang} for compact immersions. We also prove a Omori-Yau maximum principle for properly immersed self-shrinkers, which improves a result in \cite{CJQ}.