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On the Poisson Integral of Step Functions and Minimal Surfaces

Published online by Cambridge University Press:  20 November 2018

Allen Weitsman
Allen Weitsman*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Inidiana 47907, U.S.A.
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Abstract

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Applications of minimal surface methods are made to obtain information about univalent harmonic mappings. In the case where the mapping arises as the Poisson integral of a step function, lower bounds for the number of zeros of the dilatation are obtained in terms of the geometry of the image.

Keywords

30C6231A0531A2049Q05harmonic mappingsdilatationminimal surfaces
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 45 , Issue 1 , 01 March 2002 , pp. 154 - 160
Copyright
Copyright © Canadian Mathematical Society 2002

References

[C] Choquet, G., Sur un type de transformation analytique généralisant la représentation conforme et définie an moyen de fonctions harmoniques. Bull. Sci.Math. 69 (1945), 156–165.Google Scholar
[CS-S] Clunie, J. and Sheil-Small, T., Harmonic univalent functions. Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 3–25.Google Scholar
[HS] Hengartner, W. and Schober, G., On the boundary behavior of orientation-preserving harmonic mappings. ComplexVariables 5 (1986), 197–208.Google Scholar
[JS] Jenkins, H. and Serrin, J., Variational problems of minimal surface type II. Boundary value problems for the minimal surface equation. Arch. Rat.Mech. Anal. 21(1965/66), 321–342.Google Scholar
[K] Kneser, H., Lösung der Aufgabe 41. Jahresber. Deutsch. Math.-Verein. 35 (1925), 123–4.Google Scholar
[N] Nitsche, J. C. C., Über ein verallgemeinertes Dirichletsches Problem für die Minimalflächengleichung und hebbare Unstetigkeiten ihrer Lösungen. Math. Ann. 158 (1965), 203–214.Google Scholar
[O] Osserman, R., A Survey of Minimal Surfaces. Dover, 1986.Google Scholar
[S] Springer, G., Introduction to Riemann Surfaces. Addison-Wesley, 1957.Google Scholar
[S-S] Sheil-Small, T., On the Fourier series of a step function. Michigan Math. J. 36 (1989), 459–475.Google Scholar
[W] Weitsman, A., On Univalent Harmonic Mappings and Minimal Surfaces. Pacific Math. J. 192 (2000), 191–200.Google Scholar