Non-existence of eternal solutions to Lagrangian mean curvature flow with non-negative Ricci curvature

  • Keita KunikawaEmail author
Original Paper


In this paper, we derive a mean curvature estimate for eternal solutions of uniformly almost calibrated Lagrangian mean curvature flow with non-negative Ricci curvature in the complex Euclidean space. As a consequence, we show a non-existence result for such eternal solutions.


Mean curvature flow Eternal solution 

Mathematics Subject Classification 2010

Primary: 53C44 Secondary: 35C06 



The author is supported by Grant-in-Aid for JSPS Fellows Number 16J01498. During the preparation of this paper the author has stayed at the Max Planck Institute for Mathematics in the Sciences, Leipzig. The author is grateful to Jürgen Jost for his hospitality and his interest. Reiko Miyaoka also gave the author helpful comments in private seminars. Finally, the author would like to thank the referees for their valuable comments which helped to improve the manuscript.


  1. 1.
    Chen, J., Li, J.: Mean curvature flow of surface in 4-manifolds. Adv. Math. 163, 287–309 (2001)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cheng, S.Y., Yau, S.T.: Differential equations on Riemannian manifolds and geometric applications. Commun. Pure Appl. Math. 28, 333–354 (1975)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Hamilton, R.S.: Harnack estimate for the mean curvature flow. J. Differ. Geom. 41(1), 215–226 (1995)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Han, X., Sun, J.: Translating solitons to symplectic mean curvature flows. Ann. Global Anal. Geom. 38, 161–169 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Haslhofer, R.: Lectures on curve shortening flow (preprint).
  6. 6.
    Kunikawa, K.: A Bernstein type theorem of ancient solutions to the mean curvature flow. Proc. Am. Math. Soc. 144, 1325–1333 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Li, P., Yau, S.T.: On the parabolic kernel of the Schödinger operator. Acta Math. 156, 153–201 (1986)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Neves, A.: Singularities of Lagrangian mean curvature flow: zero-Maslov class case. Invent. Math. 168, 449–484 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Neves, A., Tian, G.: Translating solutions to Lagrangian mean curvature flow. Trans. Am. Math. Soc. 365(11), 5655–5680 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Souplet, P., Zhang, Q.S.: Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull. Lond. Math. Soc. 38, 1045–1053 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Smoczyk, K.: A canonical way to deform a Lagrangian submanifold. arXiv:dg-ga/9605005v2 (1996)
  12. 12.
    Smoczyk, K.: Harnack inequality for the Lagrangian mean curvature flow. Calc. Var. Partial Differ. Equ. 8(3), 247–258 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Smoczyk, K.: Der Lagrangesche mittlere Krümmungsfluss (The Lagrangian mean curvature flow). Univ. Leipzig (Habil.-Schr.), Leipzig (2000)zbMATHGoogle Scholar
  14. 14.
    Sun, J.: A gap theorem for translating solitons to Lagrangian mean curvature flow. Differ. Geom. Appl. 31, 568–576 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wang, M.: Liouville theorems for the ancient solution of heat flows. Proc. Am. Math. Soc. 139(10), 3491–3496 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wang, M.T.: Mean curvature flow of surfaces in Einstein four manifolds. J. Differ. Geom. 57(2), 301–338 (2001)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Advanced Institute for Materials ResearchTohoku UniversitySendaiJapan

Personalised recommendations