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The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3183–3195 | Cite as

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

  • John Man Shun Ma
Article
  • 136 Downloads

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).

Keywords

Mean curvature flow Omori–Yau maximum principle Self-shrinkers Gauss map 

Mathematics Subject Classification

53C44 

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Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityPiscatawayUSA

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