Skip to main content
SpringerLink
Log in

Higher Codimension Minimal Submanifold with Isolated Singularity

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

We study the existence of higher codimension minimal submanifold with isolated singularity but is not a cone. We generalize the fix point method in Caffarelli et al. (Manuscripta Math 48(1–3):1–18, 1984) to higher codimension setting and show the existence of non-conical higher codimension minimal submanifold (with boundary) with isolated singularity.

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

Similar content being viewed by others

References

  1. Almgren, F.: Q valued functions minimizing dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two. Princeton, NJ, (1984)

  2. Bombieri, E., De Giorgi, E., Giusti, E.: Minimal cones and the Bernstein problem. Invent. Math. 7, 243–268 (1969)

    Article  MathSciNet  Google Scholar 

  3. Caffarelli, L., Hardt, R., Simon, L.: Minimal surfaces with isolated singularities. Manuscripta Math. 48(1–3), 1–18 (1984)

    Article  MathSciNet  Google Scholar 

  4. Gilbarg, D., Trudinger, N.S., Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). (Reprint of the 1998 edition)

  5. Harvey, R., Blaine Lawson, H., Jr.: Calibrated geometries. Acta Math. 148, 47–157 (1982)

    Article  MathSciNet  Google Scholar 

  6. Hardt, R., Simon, L.: Area minimizing hypersurfaces with isolated singularities. J. R. Angew. Math. 1985(362), 102–129 (1985)

    Article  MathSciNet  Google Scholar 

  7. Lawson, H.B., Jr., Osserman, R.: Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system. Acta Math. 139(1–2), 1–17 (1977)

    Article  MathSciNet  Google Scholar 

  8. Simons, J.: Minimal varieties in Riemannian manifolds. Ann. Math. 2(88), 62–105 (1968)

    Article  MathSciNet  Google Scholar 

  9. Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematical Analysis, vol. 3. Australian National University. Australian National University, Centre for Mathematical Analysis, Canberra (1983)

  10. Smale, N.: A bridge principle for minimal and constant mean curvature submanifolds of \({ R}^N\). Invent. Math. 90(3), 505–549 (1987)

    Article  MathSciNet  Google Scholar 

  11. Smale, N.: Minimal hypersurfaces with many isolated singularities. Ann. Math. 130(3), 603–642 (1989)

    Article  MathSciNet  Google Scholar 

  12. Smale, N.: An equivariant construction of minimal surfaces with nontrivial singular sets. Indiana Univ. Math. J. 40(2), 595–616 (1991)

    Article  MathSciNet  Google Scholar 

  13. Smale, N.: Geometric P.D.E.s with isolated singularities. J. Reine Angew. Math. 440, 1–41 (1993)

    MathSciNet  MATH  Google Scholar 

  14. Xin, Y.: Minimal submanifolds and related topics, volume 16 of Nankai Tracts in Mathematics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2019). Second edition of [MR2035469]

  15. Xiaowei, X., Yang, L., Zhang, Y.: Dirichlet boundary values on Euclidean balls with infinitely many solutions for the minimal surface system. J. Math. Pures Appl. 129, 266–300 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is supported by 108-2115-M-002-009-MY3. The author would like to thank Yng-Ing Lee from National Taiwan University for discussions. The author would also like to thank the referee for valuable suggestion.

Author information

Authors and Affiliations

  1. National Taiwan University, Taipei, Taiwan

    Yuan Shyong Ooi

Authors
  1. Yuan Shyong Ooi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuan Shyong Ooi.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ooi, Y.S. Higher Codimension Minimal Submanifold with Isolated Singularity. J Geom Anal 32, 164 (2022). https://doi.org/10.1007/s12220-022-00904-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-022-00904-4

Keywords

Search

Navigation