The Journal of Geometric Analysis

, Volume 1, Issue 4, pp 329–338 | Cite as

Minimal cones on hypercubes

  • Kenneth A. Brakke


It is shown that in dimension greater than four, the minimal area hypersurface separating the faces of a hypercube is the cone over the edges of the hypercube. This constrasts with the cases of two and three dimensions, where the cone is not minimal. For example, a soap film on a cubical frame has a small rounded square in the center. In dimensions over 6, the cone is minimal even if the area separating opposite faces is given zero weight. The proof uses the maximal flow problem that is dual to the minimal surface problem.

Math Subject Classification

49Q05 49Q20 49N15 

Key Words and Phrases

Calculus of variations calibrations hypercube minimal cones minimal surfaces 


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  1. [BDG]
    Bombieri, E., DeGiorgi, E., and Guisti, E. Minimal cones and the Bernstein theorem. Invent. Math.7, 243–269 (1969).MATHCrossRefMathSciNetGoogle Scholar
  2. [BK1]
    Brakke, K. Minimal surfaces and maximal flows. In preparation.Google Scholar
  3. [BK2]
    Brakke, K. Surface Evolver program. Source code and documentation available via anonymous ftp from in the pub directory as evolver.tar. Z. Code is in C, runs on many systems, and should be easily portable to any C system. A printed version of the documentation is available asSurface Evolver Manual, Research Report GCG 31 (1991) from The Geometry Center, 1300 South Second Street, Minneapolis, MN 55454 USA.Google Scholar
  4. [FH1]
    Federer, H. Geometric Measure Theory. New York: Springer-Verlag 1969.MATHGoogle Scholar
  5. [FH2]
    Federer, H. Real flat chains, cochains and variational problems. Indiana Univ. Math. J.24 351–407 (1974).MATHCrossRefMathSciNetGoogle Scholar
  6. [LM]
    Lawlor, G., and Frank M. Minimizing cones and networks: Immiscible fluids, norms, and calibrations. Preliminary version.Google Scholar
  7. [TJ]
    Taylor, J. The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces. Ann. Math.103, 489–539 (1976).CrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 1991

Authors and Affiliations

  • Kenneth A. Brakke
    • 1
  1. 1.Susquehanna UniversityMathematics DepartmentSelinsgroveUSA

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