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Explicit Determination in \(\mathbb {R} ^{N}\) of \((N-1)\)-Dimensional Area Minimizing Surfaces with Arbitrary Boundaries

  • Harold R. ParksEmail author
  • Jon T. Pitts
Article
  • 3 Downloads

Abstract

Let \(N\ge 3\) be an integer and B be a smooth, compact, oriented, \((N-2)\)-dimensional boundary in \(\mathbb {R} ^{N}\). In 1960, Federer and Fleming (Ann Math 72:458–520, 1960) proved that there is an \((N-1)\)-dimensional integral current spanning surface of least area. The proof was by compactness methods and non-constructive. Thus, it is a question of long standing whether there is a numerical algorithm that will closely approximate the area-minimizing surface. The principal result of this paper is an algorithm that solves this problem—with the proviso that since one cannot guarantee the uniqueness of the area-minimizing surface with a particular given boundary, one must be willing to alter the boundary slightly, but by no more than a small amount that can be limited in advance. Our algorithm is currently theoretical rather than practical. Specifically, given a neighborhood U around B in \(\mathbb {R} ^{N}\) and a tolerance \(\epsilon >0\), we prove that one can explicitly compute in finite time an \((N-1)\)-dimensional integral current T with the following approximation requirements:
  1. (1)

    \({\text {spt}}(\partial T)\subset U\).

     
  2. (2)

    B and \(\partial T\) are within distance \(\epsilon \) in the Hausdorff distance.

     
  3. (3)

    B and \(\partial T\) are within distance \(\epsilon \) in the flat norm distance.

     
  4. (4)

    \(\mathbb {M} (T)<\epsilon +\inf \{\mathbb {M} (S):\partial S=B\}\).

     
  5. (5)

    Every area-minimizing current R with \(\partial R=\partial T\) is within flat norm distance \(\epsilon \) of T.

     

Keywords

Area-minimizing surfaces Mass-minimizing currents Codimension one surfaces Flat norm approximation Multigrid approximation Finite time algorithm 

Mathematics Subject Classification

49Q15 49Q20 49Q05 

Notes

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Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Department of MathematicsOregon State UniversityCorvallisUSA
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA

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