Skip to main content
SpringerLink
Log in

On the gluing construction of translating solitons to Lagrangian mean curvature flow

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We construct Lagrangian translating solitons by desingularizing the intersection points between Lagrangian Grim Reaper cylinders with the same phase using special Lagrangian Lawlor necks. The resulting Lagrangian translating solitons could have arbitrarily many ends and non-contractible loops.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Aubin, T.: Some Nonlinear Problems in Riemannian Geometry. Springer Monographs in Mathematics. Springer, Berlin (1998). https://doi.org/10.1007/978-3-662-13006-3

    Book  Google Scholar 

  2. Biquard, O., Macbeth, H.: Steady kähler-ricci solitons on crepant resolutions of finite quotients of \(\mathbb{C} ^{n}\) (2017)

  3. Castro, I., Lerma, A.M.: Translating solitons for Lagrangian mean curvature flow in complex Euclidean plane. Int. J. Math. 23(10), 1250101 (2012). https://doi.org/10.1142/S0129167X12501017

    Article  MathSciNet  MATH  Google Scholar 

  4. Conlon, R.J., Deruelle, A.: Steady gradient Kähler–Ricci solitons on crepant resolutions of Calabi–Yau cones (2020)

  5. Dávila, J., del Pino, M., Nguyen, X.H.: Finite topology self-translating surfaces for the mean curvature flow in \(\mathbb{R} ^{3}\). Adv. Math. 320, 674–729 (2017). https://doi.org/10.1016/j.aim.2017.09.014

    Article  MathSciNet  MATH  Google Scholar 

  6. Harvey, R., Lawson, H.B., Jr.: Calibrated geometries. Acta Math. 148, 47–157 (1982). https://doi.org/10.1007/BF02392726

    Article  MathSciNet  MATH  Google Scholar 

  7. Imagi, Y., Joyce, D., dos Santos, O.J.: Uniqueness, results for special Lagrangians and Lagrangian mean curvature flow expanders in \(\mathbb{C} ^m\). Duke Math. J. 165(5), 847–933 (2016). https://doi.org/10.1215/00127094-3167275

    Article  MathSciNet  MATH  Google Scholar 

  8. Joyce, D.: Special Lagrangian submanifolds with isolated conical singularities. I. Regularity. Ann. Glob. Anal. Geom. 25(3), 201–251 (2004). https://doi.org/10.1023/B:AGAG.0000023229.72953.57

    Article  MathSciNet  MATH  Google Scholar 

  9. Joyce, D.: Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case. Ann. Glob. Anal. Geom. 26(1), 1–58 (2004). https://doi.org/10.1023/B:AGAG.0000023231.31950.cc

    Article  MathSciNet  MATH  Google Scholar 

  10. Joyce, D.: Conjectures on Bridgeland stability for Fukaya categories of Calabi–Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow. EMS Surv. Math. Sci. 2(1), 1–62 (2015). https://doi.org/10.4171/EMSS/8

    Article  MathSciNet  MATH  Google Scholar 

  11. Joyce, D., Lee, Y.-I., Tsui, M.-P.: Self-similar solutions and translating solitons for Lagrangian mean curvature flow. J. Differ. Geom. 84(1), 127–161 (2010)

    Article  MathSciNet  Google Scholar 

  12. Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Sobolev Spaces. Graduate Studies in Mathematics. American Mathematical Society, New York (2008)

    Book  Google Scholar 

  13. Lawlor, G.: The angle criterion. Invent. Math. 95(2), 437–446 (1989). https://doi.org/10.1007/BF01393905

    Article  MathSciNet  MATH  Google Scholar 

  14. Lockhart, R.B., McOwen, R.C.: Elliptic differential operators on noncompact manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12(3), 409–447 (1985)

    MathSciNet  MATH  Google Scholar 

  15. Marshall, S.: Deformations of special Lagrangian submanifolds. Ph.D. thesis (2002)

  16. Neves, A.: Singularities of Lagrangian mean curvature flow: zero-Maslov class case. Invent. Math. 168(3), 449–484 (2007). https://doi.org/10.1007/s00222-007-0036-3

    Article  MathSciNet  MATH  Google Scholar 

  17. Nguyen, X.H.: Translating tridents. Commun. Partial Differ. Equ. 34(3), 257–280 (2009). https://doi.org/10.1080/03605300902768685

    Article  MathSciNet  MATH  Google Scholar 

  18. Nguyen, X.H.: Complete embedded self-translating surfaces under mean curvature flow. J. Geom. Anal. 23(3), 1379–1426 (2013). https://doi.org/10.1007/s12220-011-9292-y

    Article  MathSciNet  MATH  Google Scholar 

  19. Nguyen, X.H.: Doubly periodic self-translating surfaces for the mean curvature flow. Geom. Dedicata 174, 177–185 (2015). https://doi.org/10.1007/s10711-014-0011-2

    Article  MathSciNet  MATH  Google Scholar 

  20. Pacini, T.: Desingularizing isolated conical singularities: uniform estimates via weighted Sobolev spaces. Commun. Anal. Geom. 21 (2010)

  21. Pacini, T., Special Lagrangian conifolds, II: gluing constructions in \(\mathbb{C} ^{m}\). In: Proceedings of the London Mathematical Society, vol. 107 (2011)

  22. Savas-Halilaj, A., Smoczyk, K.: Lagrangian mean curvature flow of Whitney spheres. arXiv e-prints arXiv:1802.06304

  23. Su, W.-B.: f-Minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials. Int. Math. Res. Not. 4, 2539–2564 (2021)

    Article  Google Scholar 

  24. Weinstein, A.: Symplectic manifolds and their Lagrangian submanifolds. Adv. Math. 6(3), 329–346 (1971)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to thank Dominic Joyce for suggesting him to think about gluing construction for Lagrangian translating solitons. He also wants to thank Yng–Ing Lee for stimulating discussions.

Author information

Authors and Affiliations

  1. Academia Sinica, Taipei, Taiwan

    Wei-Bo Su

Authors
  1. Wei-Bo Su
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei-Bo Su.

Additional information

Communicated by A. Neves.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Su, WB. On the gluing construction of translating solitons to Lagrangian mean curvature flow. Calc. Var. 61, 218 (2022). https://doi.org/10.1007/s00526-022-02337-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-022-02337-x

Mathematics Subject Classification

Search

Navigation