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


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).


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

Mathematics Subject Classification



  1. 1.
    Alías, L., Mastrolia, P., Rigoli, M.: Maximum Principles and Geometric Applications. Springer Monographs in Mathematics. Springer, New York (2016). ISBN 978-3-319-24335-1Google Scholar
  2. 2.
    Chen, Q., Jost, J., Qiu, H.: Omori-Yau maximum principles, V-harmonic maps and their geometric applications. Ann. Glob. Anal. Geom. 46, 259–279 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, Q., Xin, Y.L.: A generalized maximum principle and its applications in geometry. Am. J. Math. 114(2), 355–366 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cheng, Q.M., Peng, Y.: Complete self-shrinkers of the mean curvature flow. Calc. Var. 52, 497 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Colding, T., Minicozzi, W.: Generic mean curvature flow I: generic singularities. Ann. Math. (2) 175(2), 755–833 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. Math. 2(130), 453–471 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Li, P., Wang, J.: Comparison theorem for Kähler manifolds and positivity of spectrum. J. Differ. Geom. 69(1), 043–074 (2005)CrossRefGoogle Scholar
  8. 8.
    Neves, A.: Singularities of Lagrangian mean curvature flow: zero-Maslov class case. Invent. math. 168, 449 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Omori, H.: Isometric immersion of Riemannian manifolds. J. Math. Soc. Jpn. 19, 205–214 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Pigola, S., Rigoli, M., Setti, A.: Maximum Principle on Riemannian Manifolds and Applications. Memoirs of the American Mathematical Society, vol. 174(822). American Mathematical Soc, Providence (2005)zbMATHGoogle Scholar
  11. 11.
    Smoczyk, K.: A canonical way to deform a Lagrangian submanifold. arXiv:dg-ga/9605005
  12. 12.
    Wang, M.T.: Gauss map of the mean curvature flow. Math. Res. Lett. 10, 287–299 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yau, S.T.: Harmonic function on complete Riemannian manifolds. Commun. Pure. Appl. Math. 28, 201–228 (1975)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityPiscatawayUSA

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