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Natural algebraic representation formulas for curves in ℂ³

100%

Gollek H.

Banach Center Publications

2002

tom 57

nr 1

109-134

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.

Algebraic representation formulas for null curves in Sl(2,ℂ)

100%

Gollek H.

Banach Center Publications

2005

tom 69

nr 1

221-242

We study curves in Sl(2,ℂ) whose tangent vectors have vanishing length with respect to the biinvariant conformal metric induced by the Killing form, so-called null curves. We establish differential invariants of them that resemble infinitesimal arc length, curvature and torsion of ordinary curves in Euclidean 3-space. We discuss various differential-algebraic representation formulas for null curves. One of them, a modification of the Bianchi-Small formula, gives an Sl(2,ℂ)-equivariant bijection between pairs of meromorphic functions and null curves. The inverse of this formula is also differential-algebraic. The other one is based on an integral formula deduced from that of R. Bryant, using certain natural differential operators on Riemannian surfaces that we introduced in [7] for differential-algebraic representation formulas of curves in ℂ³. We demonstrate some commands of a Mathematica package that resulted from our investigations, containing algebraic and graphical utilities to handle null curves, their invariants, representation formulas and associated surfaces of constant mean curvature 1 in ℍ³, taking into consideration several models of ℍ³.

Ograniczanie wyników

2 Banach Center Publications

2 Gollek H.

1 2005

1 2002

Wyniki wyszukiwania

Natural algebraic representation formulas for curves in ℂ³

Algebraic representation formulas for null curves in Sl(2,ℂ)