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Transactions of the American Mathematical Society

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On the singular structure of three-dimensional, area-minimizing surfaces


Author: Frank Morgan
Journal: Trans. Amer. Math. Soc. 276 (1983), 137-143
MSC: Primary 49F20; Secondary 53A10
DOI: https://doi.org/10.1090/S0002-9947-1983-0684498-4
MathSciNet review: 684498
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Abstract: A sufficient condition is given for the union of two three-dimensional planes through the origin in ${{\mathbf {R}}^n}$ to be area-minimizing. The condition is in terms of the three angles $0 \leqslant {\gamma _1} \leqslant {\gamma _2} \leqslant {\gamma _3}$ which characterize the geometric relationship between the planes. If ${\gamma _3} \leqslant {\gamma _1} + {\gamma _2}$, the union of the planes is area-minimizing.


References [Enhancements On Off] (What's this?)

    F. J. Almgren, Jr., Multiple valued functions minimizing Dirichlet’s integral and the regularity of mass minimizing integral currents (in preparation).
  • Robert L. Bryant, Submanifolds and special structures on the octonians, J. Differential Geometry 17 (1982), no. 2, 185–232. MR 664494
  • Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
  • Reese Harvey and Blaine Lawson, Student reminiscences of Kodaira at Stanford, Asian J. Math. 4 (2000), no. 1, iv. Kodaira’s issue. MR 1803726
  • Frank Morgan, On the singular structure of two-dimensional area minimizing surfaces in ${\bf R}^{n}$, Math. Ann. 261 (1982), no. 1, 101–110. MR 675210, DOI https://doi.org/10.1007/BF01456413
  • Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), no. 3, 489–539. MR 428181, DOI https://doi.org/10.2307/1970949

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Keywords: Area-minimizing, mass-minimizing, singular structure
Article copyright: © Copyright 1983 American Mathematical Society