Skip to main content
SpringerLink
Log in

Rigidity Results on Lagrangian and Symplectic Translating Solitons

  • Published:
Communications in Mathematics and Statistics Aims and scope Submit manuscript

Abstract

In this short note, we prove that an almost calibrated Lagrangian translating soliton must be a plane if it has weighted integrable mean curvature vector or weighted quadratic area growth. Similar results are also true for symplectic translating solitons.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Chen, J., Li, J.: Mean curvature flow of surfaces in 4-manifolds. Adv. Math. 163, 287–309 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen, J., Li, J.: Singularity of mean curvature flow of Lagrangian submanifolds. Invent. Math. 156(1), 25–51 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Hamilton, R.S.: Harnack estimate for the mean curvature flow. J. Differ. Geom. 41, 215–226 (1995)

    MATH  Google Scholar 

  4. Han, X., Li, J.: Translating solitons to symplectic and Lagrangian mean curvature flows. Int. J. Math. 20(4), 443–458 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Han, X., Sun, J.: Translating solitons to symplectic mean curvature flows. Ann. Glob. Anal. Geom. 38, 161–169 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Huisken, G., Sinestrari, C.: Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183(1), 45–70 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Sun, J.: A gap theorem for translating solitons to Lagrangian mean curvature flow. Differ. Geom. Appl. 31, 568–576 (2013)

    Article  Google Scholar 

  10. Sun, J.: Mean curvature decay in symplectic and Lagrangian translating solitons. Geom. Dedicata 172, 207–215 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  11. Wang, M.-T.: Mean curvature flow of surfaces in Einstein four manifolds. J. Differ. Geom. 57, 301–338 (2001)

    MATH  Google Scholar 

  12. White, B.: The nature of singularities in mean curvature flow of mean-convex sets. J. Am. Math. Soc. 16, 123–138 (2003)

    Article  MATH  Google Scholar 

  13. Xin, Y. L.: The translating soliton of mean curvature flow, preprint, arXiv:1410.5063v1

Download references

Acknowledgments

The author was supported by NSF in China, No. 11401440.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s Republic of China

    Jun Sun

Authors
  1. Jun Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jun Sun.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, J. Rigidity Results on Lagrangian and Symplectic Translating Solitons. Commun. Math. Stat. 3, 63–68 (2015). https://doi.org/10.1007/s40304-015-0052-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40304-015-0052-3

Keywords

Mathematics Subject Classification

Search

Navigation