Skip to main content

A note on the singular set of area-minimizing hypersurfaces

Abstract

We prove an isoperimetric-type bound on the \((n-7)\)-dimensional measure of the singular set for a large class of area-minimizing n-dimensional hypersurfaces, in terms of the geometry of their boundary.

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

References

  1. 1.

    Allard, W.: On the first variation of a varifold. Ann. Math. 2(95), 417–491 (1972)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Almgren Jr., F.J.: Optimal isoperimetric inequalities. Indiana Univ. Math. J. 35, 451–547 (1986)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Almgren, F.J., Lieb, Elliott H.: Singularities of energy minimizing maps from the ball to the sphere: examples, counterexamples, and bounds. Ann. Math. 128(3), 483–530 (1988)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bombieri, E.: Regularity theory for almost minimal currents. Arch. Rational Mech. Anal. 78, 99–130 (1982)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Duzaar, F., Steffan, K.: Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals. J. Reine Angew. Math. 546, 73–138 (2002)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Edelen, N., Engelstein, M.: Quantitative stratification for some free-boundary problems. Trans. Am. Math. Soc. (2017)

  7. 7.

    Edelen, N.: Notes on a measure theoretic version of Naber–Valtorta’s rectifiability theorem. http://math.mit.edu/~edelen/general-nv.pdf. Accessed 10 Jan 2019

  8. 8.

    Gruter, M.: Optimal regularity for codimension one minimal surfaces with a free-boundary. Manuscr. Math. 58, 295–343 (1987)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Gruter, M., Jost, J.: Allard type regularity results for varifolds with free boundaries. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13, 129–169 (1986)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Hardt, R., Simon, L.: Boundary regularity and embedded solutions for the oriented plateau problem. Ann. Math. 2(110), 439–486 (1979)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Mazowiecka, K., Miskiewicz, M., Schikorra, A.: On the size of the singular set of minimizing harmonic maps into the sphere in dimension three (2018). arXiv:1811.00515

  12. 12.

    Mazowiecka, K, Miskiewicz, M., Schikorra, A.: On the size of the singular set of minimizing harmonic maps into a \(2\) sphere in dimension four and higher (2018). arXiv:1902.03161

  13. 13.

    Naber, A., Valtorta, D.: The singular structure and regularity of stationary and minimizing varifolds (2015). arXiv:1505.03428

  14. 14.

    Naber, A., Valtorta, D.: Rectifiable-reifenberg and the regularity of stationary and minimizing harmonic maps. Ann. Math. 2(185), 131–227 (2017)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Perez, J.: Stable embedded minimal surfaces bounded by a straight line. Calc. Variations Partial Differ Equ 29, 267–279 (2007)

    MathSciNet  Article  Google Scholar 

  16. 16.

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

  17. 17.

    Simon, L.: Rectifiability of the singular set of energy minimizing maps. Calc. Variations Partial Differ. Equ. 3, 1–65 (1995)

    MathSciNet  Article  Google Scholar 

  18. 18.

    White, B.: Half of Enneper’s surface minimizes area. In: Jost, J. (ed.) Geometric Analysis and the Calculus of Variations for Stefan Hildebrandt, pp. 361–367. International Press, Somerville (1996)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Nick Edelen.

Additional information

Publisher's Note

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

Communicated by C. de Lellis.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Edelen, N. A note on the singular set of area-minimizing hypersurfaces. Calc. Var. 59, 18 (2020). https://doi.org/10.1007/s00526-019-1660-7

Download citation

Mathematics Subject Classification

  • 53
  • 49
  • 28