The geometric structure of solutions to the two-valued minimal surface equation

  • Leobardo RosalesEmail author


Recently, Simon and Wickramasekera (J Differ Geom 75:143–173, 2007) introduced a PDE method for producing examples of stable branched minimal immersions in \({\mathbb{R}^{3}}\). This method produces two-valued functions u over the punctured unit disk in \({\mathbb{R}^{2}}\) so that either u cannot be extended continuously across the origin, or G the two-valued graph of u is a C 1,α stable branched immersed minimal surface. The present work gives a more complete description of these two-valued graphs G in case a discontinuity does occur, and as a result, we produce more examples of C 1,α stable branched immersed minimal surfaces, with a certain evenness symmetry.

Mathematics Subject Classification (2000)

49Q05 49Q15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allard W.K.: On the first variation of a varifold-boundary behavior. Ann. Math. 101, 418–446 (1975)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Allard W.K., Almgren F.J. Jr.: The structure of stationary one dimensional varifolds with positive density. Invent. Math. 34, 83–97 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Finn R.: Isolated singularities of solutions of non-linear partial differential equations. Trans. Am. Math. Soc. 75, 385–404 (1953)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Schoen R., Simon L.: Regularity of stable minimal hypersurfaces. Commun. Pure Appl. Math. 34, 741–797 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Schoen R., Simon L.: Regularity of simply connected surfaces with quasiconformal Gauss map. Ann. Math. Stud. 103, 127–145 (1983)MathSciNetGoogle Scholar
  6. 6.
    Simon L.: A Hölder estimate for quasiconformal maps between surfaces in Euclidean space. Acta Math. 139, 19–51 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Simon, L.: Lectures on geometric measure theory. Centre for Mathematical Analysis, Australian National University, Australia (1984)Google Scholar
  8. 8.
    Simon L., Wickramasekera N.: Stable branched minimal immersions with prescribed boundary. J. Differ. Geom. 75, 143–173 (2007)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Wickramasekera N.: A regularity and compactness theory for immersed stable minimal hypersurfaces of multiplicity at most 2. J. Differ. Geom. 80, 79–173 (2008)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Rice UniversityHoustonUSA

Personalised recommendations