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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the area of harmonic surfaces
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by Michael Beeson
Proc. Amer. Math. Soc. 69 (1978), 143-147
DOI: https://doi.org/10.1090/S0002-9939-1978-0482818-0

Abstract:

We prove that if the sequence ${S_n}$ of harmonic surfaces converges to the harmonic surface S, and if the boundary of ${S_n}$ is a rectifiable Jordan curve ${C_n}$, whose length is uniformly bounded by L, then the area $A({S_n})$ converges to $A(S)$. This solves an old problem, several special cases of which have been solved in the literature.
References
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  • Michael J. Beeson, Principles of continuous choice and continuity of functions in formal systems for constructive mathematics, Ann. Math. Logic 12 (1977), no. 3, 249–322. MR 485259, DOI 10.1016/S0003-4843(77)80003-X
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  • Erhard Heinz, Unstable surfaces of constant mean curvature, Arch. Rational Mech. Anal. 38 (1970), 257–267. MR 262942, DOI 10.1007/BF00281523
  • Marston Morse and C. Tompkins, The continuity of the area of harmonic surfaces as a function of the boundary representations, Amer. J. Math. 63 (1941), 825–838. MR 6027, DOI 10.2307/2371624
  • Johannes C. C. Nitsche, Vorlesungen über Minimalflächen, Die Grundlehren der mathematischen Wissenschaften, Band 199, Springer-Verlag, Berlin-New York, 1975 (German). MR 0448224
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Bibliographic Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 69 (1978), 143-147
  • MSC: Primary 58E15; Secondary 49F25, 53A10
  • DOI: https://doi.org/10.1090/S0002-9939-1978-0482818-0
  • MathSciNet review: 0482818