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).
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Ma, J.M.S. Parabolic Omori–Yau Maximum Principle for Mean Curvature Flow and Some Applications. J Geom Anal 28, 3183–3195 (2018). https://doi.org/10.1007/s12220-017-9954-5
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DOI: https://doi.org/10.1007/s12220-017-9954-5