On the area of harmonic surfaces

Author:
Michael Beeson

Journal:
Proc. Amer. Math. Soc. **69** (1978), 143-147

MSC:
Primary 58E15; Secondary 49F25, 53A10

MathSciNet review:
0482818

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Abstract: We prove that if the sequence of harmonic surfaces converges to the harmonic surface *S*, and if the boundary of is a rectifiable Jordan curve , whose length is uniformly bounded by *L*, then the area converges to . This solves an old problem, several special cases of which have been solved in the literature.

**[Ba]**Emilio Baiada,*L’area delle superficie armoniche quale funzione delle rappresentazioni del contorno*, Rivista Mat. Univ. Parma**2**(1951), 315–330 (Italian). MR**0047125****[B1]**M. Beeson,*Plateau's problem and constructive mathematics*(to appear).**[B2]**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**0485259****[C]**R. Courant,*Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces*, Interscience Publishers, Inc., New York, N.Y., 1950. Appendix by M. Schiffer. MR**0036317****[H]**Erhard Heinz,*Unstable surfaces of constant mean curvature*, Arch. Rational Mech. Anal.**38**(1970), 257–267. MR**0262942****[M-T]**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**0006027****[N]**Johannes C. C. Nitsche,*Vorlesungen über Minimalflächen*, Springer-Verlag, Berlin-New York, 1975 (German). Die Grundlehren der mathematischen Wissenschaften, Band 199. MR**0448224****[S]**Max Shiffman,*Unstable minimal surfaces with any rectifiable boundary*, Proc. Nat. Acad. Sci. U. S. A.**28**(1942), 103–108. MR**0006029**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1978-0482818-0

Article copyright:
© Copyright 1978
American Mathematical Society