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Björling Curve

Let alpha(z),gamma(z):(a,b)->R^3 be curves such that |gamma|=1 and alpha·gamma=0, and suppose that alpha and gamma have holomorphic extensions alpha,gamma:(a,b)×(c,d)->C^3 such that |gamma|=1 and alpha·gamma=0 also for z in (a,b)×(c,d). Fix z_0 in (a,b)×(c,d). Then the Björling curve, defined by

B(z)=alpha(z)-iint_(z_0)^zgamma(z)xalpha^'(z)dz,

is a minimal curve (Gray 1997, p. 762).

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References

Björling, E. G. "In integrationem aequationis derivatarum partialum superficiei, cujus in puncto, unoquoque principales ambo radii curvedinis aequales sunt signoque contrario." Arch. Math. Phys. 4, 290-315, 1844.Dierkes, U.; Hildebrand, S.; Küster, A.; and Wohlrab, O. Minimal Surfaces I: Boundary Value Problems. New York: Springer-Verlag, pp. 120-135, 1992.Gray, A. "Minimal Surfaces via Björling's Formula." Ch. 33 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 761-772, 1997.Nitsche, J. C. C. Lectures on Minimal Surfaces, Vol. 1: Introduction, Fundamentals, Geometry and Basic Boundary Value Problems. Cambridge, England: Cambridge University Press, pp. 139-145, 1989.Schwarz, H. A. Gesammelte Mathematische Abhandlungen, Vols. 1-2. New York: Chelsea, pp. 179-189, 1972.

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Björling Curve

Cite this as:

Weisstein, Eric W. "Björling Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BjoerlingCurve.html

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