Natural algebraic representation formulas for curves in ℂ³

Hubert Gollek. "Natural algebraic representation formulas for curves in ℂ³." Banach Center Publications 57.1 (2002): 109-134. <http://eudml.org/doc/281946>.

@article{HubertGollek2002,
abstract = {We consider several explicit examples of solutions of the differential equation Φ₁’²(z) + Φ₂’²(z) + Φ₃’²(z) = d²(z) of meromorphic curves in ℂ³ with preset infinitesimal arclength function d(z) by nonlinear differential operators of the form (f,h,d) → V(f,h,d), V = (Φ₁,Φ₂,Φ₃), whose arguments are triples consisting of a meromorphic function f, a meromorphic vector field h, and a meromorphic differential 1-form d on an open set U ⊂ ℂ or, more general, on a Riemann surface Σ. Most of them are natural in the sense of ’natural operators’ as considered in [8]. The special case d(z) = 0 related to minimal curves in ℂ³ and minimal surfaces in ℝ³ is of main interest. We start with the invariant construction of a sequence $V^\{(n)\}$ of natural operators assigning to each pair (f,h) consisting of a meromorphic function f and a meromorphic vector field h on Σ a minimal curve $V^\{(n)\}(f,h): Σ → ℂ³$. The operator $V^\{(3)\}$ is bijective and equivariant on a generic set of pairs (f,h). Algebraic representation formulas of minimal surfaces that arise from evolutes and caustics of curves in ℝ² in connection with the Björling representation formula are discussed. We apply the computer algebra system Mathematica to handle big algebraic expressions describing these differential operators and to provide graphical examples of minimal surfaces produced by them.},
author = {Hubert Gollek},
journal = {Banach Center Publications},
keywords = {natural algebraic representation formulas; minimal curves in ; minimal surfaces in ; evolutes and caustics of curves in ; Björling representation formula; helicoids},
language = {eng},
number = {1},
pages = {109-134},
title = {Natural algebraic representation formulas for curves in ℂ³},
url = {http://eudml.org/doc/281946},
volume = {57},
year = {2002},
}

TY - JOUR
AU - Hubert Gollek
TI - Natural algebraic representation formulas for curves in ℂ³
JO - Banach Center Publications
PY - 2002
VL - 57
IS - 1
SP - 109
EP - 134
AB - We consider several explicit examples of solutions of the differential equation Φ₁’²(z) + Φ₂’²(z) + Φ₃’²(z) = d²(z) of meromorphic curves in ℂ³ with preset infinitesimal arclength function d(z) by nonlinear differential operators of the form (f,h,d) → V(f,h,d), V = (Φ₁,Φ₂,Φ₃), whose arguments are triples consisting of a meromorphic function f, a meromorphic vector field h, and a meromorphic differential 1-form d on an open set U ⊂ ℂ or, more general, on a Riemann surface Σ. Most of them are natural in the sense of ’natural operators’ as considered in [8]. The special case d(z) = 0 related to minimal curves in ℂ³ and minimal surfaces in ℝ³ is of main interest. We start with the invariant construction of a sequence $V^{(n)}$ of natural operators assigning to each pair (f,h) consisting of a meromorphic function f and a meromorphic vector field h on Σ a minimal curve $V^{(n)}(f,h): Σ → ℂ³$. The operator $V^{(3)}$ is bijective and equivariant on a generic set of pairs (f,h). Algebraic representation formulas of minimal surfaces that arise from evolutes and caustics of curves in ℝ² in connection with the Björling representation formula are discussed. We apply the computer algebra system Mathematica to handle big algebraic expressions describing these differential operators and to provide graphical examples of minimal surfaces produced by them.
LA - eng
KW - natural algebraic representation formulas; minimal curves in ; minimal surfaces in ; evolutes and caustics of curves in ; Björling representation formula; helicoids
UR - http://eudml.org/doc/281946
ER -