Advertisement

The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 733–784 | Cite as

Co-dimension One Area-Minimizing Currents with \(C^{1,\alpha }\) Tangentially Immersed Boundary

  • Leobardo RosalesEmail author
Article
  • 36 Downloads

Abstract

We introduce and study co-dimension one area-minimizing locally rectifiable currents T with \(C^{1,\alpha }\) tangentially immersed boundary: \(\partial T\) is locally a finite sum of orientable co-dimension two \(C^{1,\alpha }\) submanifolds which only intersect tangentially with equal orientation. We show that any such T is supported in a smooth hypersurface near any point on the support of \(\partial T\) where T has tangent cone which is a hyperplane with constant orientation but non-constant multiplicity. We also introduce and study co-dimensional one area-minimizing locally rectifiable currents T with boundary having co-oriented mean curvature: \(\partial T\) has generalized mean curvature \(H_{\partial T} = h \nu _{T}\) with h a real-valued function and \(\nu _{T}\) the generalized outward pointing unit normal of \(\partial T\) with respect to T.

Keywords

Currents Area-minimizing Boundary regularity 

Mathematics Subject Classification

28A75 49Q05 49Q15 

Notes

Acknowledgements

This work was partly supported by the associate membership program of the Korea Institute for Advanced Study. We wish to thank KIAS, and in particular Dr. Hojoo Lee, for their continued support.

References

  1. 1.
    Allard, W.K.: On the first variation of a varifold-boundary behavior. Ann. Math. 95, 417–491 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bourni, T.: Allard-type boundary regularity for \(C^{1,\alpha }\) boundaries. Adv. Calc. Var. 9, 143–161 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brothers, J.E.: Existence and structure of tangent cones at the boundary of an area-minimizing integral current. Indiana Univ. Math. J. 26, 1027–1044 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ecker, K.: Area-minimizing integral currents with movable boundary parts of prescribed mass. Ann. l’Inst. Henri Poincare (C) Non Linear Anal. 6, 261–293 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)zbMATHGoogle Scholar
  6. 6.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)CrossRefzbMATHGoogle Scholar
  7. 7.
    Hardt, R., Simon, L.: Boundary regularity and embedded solutions for the oriented Plateau problem. Ann. Math. 110, 439–483 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966)zbMATHGoogle Scholar
  9. 9.
    Pilz, R.: On the thread problem for minimal surfaces. Calc. Var. Partial Differ. Equ. 5, 117–136 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rosales, L.: The geometric structure of solutions to the two-valued minimal surface equation. Calc. Var. Partial Differ. Equ. 39, 59–84 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Rosales, L.: Discontinuous solutions to the two-valued minimal surface equation. Adv. Calc. Var. 4, 363–395 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Rosales, L.: The c-isoperimetric mass of currents and the c-Plateau problem. J. Geom. Anal. 25, 471–511 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rosales, L.: The \(q\)-valued minimal surface equation. Houston J. Math. 41, 749–765 (2015)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Rosales, L.: Partial boundary regularity for co-dimension one area-minimizing currents at immersed \(C^{1,\alpha }\) tangential boundary points (2015). arXiv:1508.04229 [math.DG]
  15. 15.
    Rosales, L.: Two-dimensional solutions to the \(c\)-Plateau problem in \(\mathbf{R}^{3}.\) Ann. Glob. Anal. Geom. (to appear) (2016)Google Scholar
  16. 16.
    Schoen, S.: Regularity of stable minimal hypersurfaces. Commun. Pure Appl. Math. 34, 741–797 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Simon, L.: Lectures on Geometric Measure Theory. Centre for Mathematical Analysis, Australian National University, Australia (1984)Google Scholar
  18. 18.
    Simon, L., Wickramasekera, N.: Stable branched minimal immersions with prescribed boundary. J. Differ. Geom. 75, 143–173 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    White, B.: Regularity of area-minimizing hypersurfaces at boundaries with multiplicity. Ann. Math. Stud. 103, 293–301 (1983)MathSciNetzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Keimyung UniversityDaeguSouth Korea
  2. 2.Korea Institute for Advanced StudySeoulSouth Korea

Personalised recommendations