Filomat 2009 Volume 23, Issue 2, Pages: 12-27
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A representation formula for curves in C3 with preset infinitesimal arc length

Gollek Hubert

We consider an algebraic representation formula for meromorphic curves in C3 with preset infinitesimal arc length, i. e., a differential operator M assigning to triples (f; h; d) of meromorphic functions meromorphic curves Ф = (φ1; φ2; φ3)T such that d is the infinitesimal arc length of Ф, in this way obtaining the complete solution of the differential equation φ021+φ022 + φ023=d2 in terms of derivatives of f; h; d only and without integrations. Computer algebra systems are an excellent tool to handle formulas of this type. We give simple Mathematica code and apply it to work out some examples, graphics as well as algebraic expressions of complex curves with special properties. For the case d = 0 of null curves, we give some graphical examples of minimal surfaces constructed in this way, showing deformations and symmetries. We give an expression for the curvature k of Ф in terms of the Schwarzian derivative of f and for the case d = 1 a simple differential relation for f and h equivalent to the condition к=1.

Keywords: curves in C3, algebraic representation formula, preset infinitesimal arc length, preset curvature, Schwarzian derivative, minimal curves, symmetries, deformations of minimal surfaces