A uniqueness theorem of complete Lagrangian translator in \(\mathbb C^2\)

Abstract

In this paper we study the complete Lagrangian translators in the complex 2-plane \(\mathbb C^2\). As the result, we obtain a uniqueness theorem showing that the plane is the only complete Lagrangian translator in \(\mathbb C^2\) with constant square norm of the second fundamental form. On the basis of this, we can prove a more general classification theorem for Lagrangian \(\xi \)-translators in \({\mathbb C}^2\). The same idea is also used to give a similar classification of Lagrangian \(\xi \)-surfaces in \({\mathbb C}^2\).

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Abresch, U., Langer, J.: The normalized curve shortening flow and homothetic solutions. J. Differ. Geom. 23, 175–196 (1986)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Altschuler, S.J., Wu, L.F.: Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc. Var. 2, 101–111 (1994)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bao, C., Shi, Y.G.: Gauss map of transliting solitons of mean curvature flow. Proc. Am. Math. Soc. 142, 1331–1344 (2014)

    Article  Google Scholar 

  4. 4.

    Castro, I., Lerma, A.M.: Hamiltonian stationary self-similar solutions for Lagrangian mean curvature flow in the complex Euclidean plane. Proc. Am. Math. Soc. 138, 1821–1832 (2010)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Castro, I., Lerma, A.M.: The Clifford torus as a self-shrinker for the Lagrangian mean curvature flow. Int. Math. Res. Not. 6, 1515–1527 (2014)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Chen, Q., Qiu, H.B.: Rigidity of self-shrinkers and translating solitons of mean curvature flows. Adv. Math. 294, 517–531 (2016)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Cheng, Q. M., Hori, H., Wei, G. X.: Complete Lagrangian self-shrinkers in \(\mathbb{R}^4\) (2018). arXiv:1802.02396v2 [math.DG]

  8. 8.

    Cheng, Q.-M., Peng, Y. J.: Complete self-shrinkers of the mean curvature flow. Calc. Var. PDE. 52(3–4), 497–506 (2015)

  9. 9.

    Cheng, X., Zhou, D.T.: Volume estimate about shrinkers. Proc. Am. Math. Soc. 141, 687–696 (2013)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Clutterbuck, J., Schnürer, O., Schulze, F.: Stability of translating solutions to mean curvature flow. Calc. Var. 29, 281–293 (2007)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Ding, Q., Xin, Y.L.: Volume growth, eigenvalue and compactness forself-shrinkers. Asian J. Math. 17(3), 443–456 (2013)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Ding, Q., Xin, Y.L.: The rigitity theorems of self-shrinkers. Trans. Am. Math. Soc. 366, 5067–5085 (2014)

    Article  Google Scholar 

  13. 13.

    Halldorsson, H.P.: Helicoidal surfaces rotating/translating under the mean curvature flow. Geom. Dedicata 162, 45–65 (2013)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Hoffman, D.A.: Surfaces of constant mean curvature in manifolds of constant curvature. J. Differ. Geom. 8, 161–176 (1973)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285–299 (1990)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Huisken, G.: Local and global behaviour of hypersurfaces moving by mean curvature. Proc. Symp. Pure Math. 54, 175–191 (1993)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Huisken, G., Sinestrari, C.: Convescity estimates for mean curvature flow and singularities of mean convex surfces. Acta. Math. 183, 45–70 (1999)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Huisken, G., Sinestrari, C.: Mean curvature flow singularities for mean convex surfaces. Calc. Vat. Partial Differ. Equ. 8, 1–14 (1999)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Li, H.Z., Wang, X.F.: New characterizations of the Clifford torus as a Lagrangian self-shrinkers. J. Geom. Anal. 27, 1393–1412 (2017)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Li, H.Z., Wei, Y.: Classification and rigidity of self-shrinkersin the mean curvature flow. J. Math. Soc. Jpn. 66, 709–734 (2014)

    Article  Google Scholar 

  21. 21.

    Li, X.X., Chang, X.F.: A rigidity theorem of \(\xi \)-submanifolds in \(\mathbb{C}^2\). Geom. Dedicata 185, 155–169 (2016). https://doi.org/10.1007/s10711-016-0173-1

    MathSciNet  Article  Google Scholar 

  22. 22.

    Li, X. X., Li, Z. P.: Variational characterizations of -submanifolds in the Eulicdean space \(\mathbb{R}^{m+p}\). Annali di Matematica Pura ed Applicata (1923 -) (2019). https://doi.org/10.1007/s10231-019-00928-8

  23. 23.

    Martín, F., Savas-Halilaj, A., Smoczyk, K.: On the topology of translating solitons of the mean curvature flow. Calc. Var. 54, 2853–2882 (2015)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Morgan, F.: Manifolds with density. Notices Am. Math. Soc. 52, 853–858 (2005)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Neves, A., Tian, G.: Translating solutions to Lagrangian mean curvature flow. Trans. Am. Math. Soc. 365(11), 5655–5680 (2013)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Pyo, J.: Compact translating solitons with non-empty planar boundary. Differ. Geom. Appl. 47, 79–85 (2016)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Shahriyari, L., Leili, Translating graphs by mean curvature flow, Ph.D., thesis, The Johns Hopkins University. p. 62(2013)

  28. 28.

    Smith, G.: On complete embedded translating solitons of the mean curvature flow that area of finite genus. Geom. Dedicata 175, 57–64 (2015)

  29. 29.

    Smoczyk, K.: Der Lagrangesche mittlere Kruemmungsfluss (The Lagrangian mean curvature flow), German Habil.-Schr., 102 S, Univ. Leipzig, Leipzig (2000)

  30. 30.

    Smoczyk, K.: Self-shrinkers of the mean curvature flow in arbitrary codimension. Int. Math. Res. Not. 48, 2983–3004 (2005)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Wang, X.-J.: Convex solutions to the mean curvature flow. Ann. Math. 173, 1185–1239 (2011)

    MathSciNet  Article  Google Scholar 

  32. 32.

    White, B.: Subsequent singularities in mean-convex mean curvature flow. Calc. Var. PDE. 54, 1457–1468 (2015)

  33. 33.

    Xin, Y.L.: Translating solitins of the mean curvature flow. Calc. Var. PDE. 54, 1995–2016 (2015)

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xingxiao Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Research supported by Foundation of Natural Sciences of China (Nos. 11671121, 11871197 and No 11971153).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, X., Liu, Y. & Qiao, R. A uniqueness theorem of complete Lagrangian translator in \(\mathbb C^2\). manuscripta math. 164, 251–265 (2021). https://doi.org/10.1007/s00229-020-01185-3

Download citation

Mathematics Subject Classification

  • 53C44
  • 53C40