Parabolic Omori–Yau Maximum Principle for Mean Curvature Flow and Some Applications

Abstract

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 uniformly bounded second fundamental forms. This generalizes the result of Wang (Math Res Lett 10:287–299, 2003) for compact immersions. We also prove a Omori–Yau maximum principle for properly immersed self-shrinkers, which improves a result in Chen et al. (Ann Glob Anal Geom 46:259–279, 2014).