Algebraic Representation Formulas for Minimal Curves in C³
by
Hubert Gollek
Humboldt-University of Berlin, Germany

Poster

A minimal curve in C³ is a meromorphic mapping \Phi:\Sigma --> C³ of a Riemann-surface \Sigma into C³ such that < d \Phi, d \Phi > =0, where < ., . > is the bilinear extension of the ordinary Euclidean scalar product of R³ to C³. Minimal curves are in close relation to minimal surfaces in R³: The set of real parts M²={Re(\Phi(p)) | p in \Sigma} subset R³ is a minimal surface with isolated singular or branch points and any local holomorphic chart z of \Sigma defines a conformal parametrization of M².

Let U subset C be an open subset and A_U be the field of meromorphic functions on U. By a (local) algebraic representation formula (ARF for short) for minimal curves we mean a (nonlinear) differential operator Re of order k mapping A_U into the space of all minimal curves defined on U. More precisely, Re(f)=F(z, f, f', f'', ... , f^(k)), where F is a 3-dimensional vector of rational functions in f, f', ... , f^(k). In a similar way we define ARF's with 2, 3 or more arguments. We would like to mention the following simple examples (of analytic and geometric origin respectively):

• the Study-formula (see [1]) and another one, that is an adaptation of a solution of the equation ((y'(x))²+(z'(x))²=1 found in [2] (page 53).

  Re1(f)(z)= æ ç ç ç è -2f + 2zf'-(z2 -1) f'' i ( 2 -2zf'+(z2+1) f'') 2(zf''-f') ö ÷ ÷ ÷ ø ,    Re2(f)(z)= æ ç ç ç è sin(z)f''-cos(z)f' cos(z)f''+sin(z)f' i(f+f'') ö ÷ ÷ ÷ ø .

• Consider the Björling type representation formula B(x, y)(z) = (x(z), y(z), -i \int_z0^z \surd{((x'(\zeta))²+(y'(\zeta))²}d \zeta), that assigns to a real analytic plane curve (x(t), y(t)) a minimal curve whose associated minimal surface intersects the x, y-plane perpendicularly in (x(t), y(t)). Combining B with the operations of assigning the evolute and the caustic respectively to a real analytic plane curve \alpha(t)=(a(t), b(t)), one obtains two ARF's B₃(a, b)(z)=(x(z), y(z), i/\kappa(z)) and B₄(a, b)(z)=([x\tilde](z), [y\tilde](z), i a(z)+i b'(z)/2\kappa(z)\surd{a'²(z)+b'²(z)}), where (x(t), y(t)) is the evolute, \kappa(t) the curvature function and ([x\tilde](t), [y\tilde](t)) the caustic of \alpha(t), i. e., the envelope of the 1-parameterfamily of lightrays, that come out as reflections at \alpha of lightrays coming in parallel to the x-axis.

A more general ARF is obtained by the following construction: Introduce a natural parameter p of a minimal curve as in [] and denote with the dash ' the derivative with respect to p. The minimal curvature (also called Study invariant) is the function defined by \kappa²(p)= < \Phi'''(p), \Phi'''(p) > . Given a minimal curve in natural parametrization such that < \Phi'''(p), \Phi'''(p) > =-1 and a meromorphic function h we define a new curve by \Delta_{\Phi, h}=(h''-h \kappa²)\Phi'-h' \Phi''+h \Phi'''. If \Phi', \Phi'', \Phi''' are linearly independent, one can show that \Phi⁽⁴⁾=\kappa' \kappa \Phi'+\kappa² \Phi''. From this observation one infers that \Delta as well as \Delta_{\Phi, h}+\Phi are again minimal curves.

Choosing \Phi as \Phi(f)(z) = \int([(i-i f²(\zeta))/(2 f(\zeta))], [(-1-f²(\zeta))/(2 f(\zeta))], [(i f(\zeta))/(f(\zeta))])d \zeta, i. e., as given by a version of the Weierstraß representation formula in terms of a meromorphic function f, we obtain a new ARF V_{f, h}=\Delta_{\Phi(f), h}.

V_{f, h} is a bijective map onto the set of parametrized minimal curves \Phi = (\phi₁, \phi₂, \phi₃) satisfying \phi'₁-i \phi'₂ =/= 0. It can be inverted without integrations by explicit algebraic operations. Moreover, V_{f, h} is a global ARF, i. e., it is the loal version of an representation formula V^\Sigma_{f, h} mapping pairs (f, h) of meromorphic functions f and meromorphic vector fields h on a Riemann-surface \Sigma into the set of all minimal curves defined on \Sigma.

The operation \Delta_{\Phi, h} can also be combined with other representation formulas to obtain other ARF's. V_{f, h} and \Delta_{\Phi, h} are involved expressions and difficult to handle. We discuss several of their properties and generalizations and provide electronic code permitting to handle them with the computer algebra system of Mathematica using ideas and techniques developed by Alfred Gray in [4].

References

[1] W. Blaschke, Vorlesungen über Differentialgeometrie.Vol. I: Elementare Differentialgeometrie, 4-th edition, Springer-Verlag, Berlin, 1945.

[2] Bryant, R.; Chern, S. S.; Gardner, Robert B.; Goldschmidt, Hubert L.; Griffiths, P. A., Exterior Differential Systems, Springer Verlag New York inc. 1991,

[3] H. Gollek, Deformations of minimal curves in C³, Proceedings of 1-st NOSONGE Conference, Warsaw, Sept 1996

[4] A. Gray, Modern Differential Geometry of Curves and Surfaces, CRC Press, 1995, 2-nd edition

http://www-irm.mathematik.hu-berlin.de/~gollek/MinSurfs

Date received: June 13, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-18.