Wydawnictwa / Banach Center Publications / Wszystkie tomy

Streszczenie

We consider several explicit examples of solutions of the differential equation $\Phi_1'^2(z)+\Phi_2'^2(z)+\Phi_3'^2(z)=d^2(z)$ of meromorphic curves in $\mathbb{C}^3$ with preset infinitesimal arclength function $d(z)$ by nonlinear differential operators of the form $(f,h,d)\to{\bf V}(f,h,d)$, ${\bf V}=\left(\Phi_1,\Phi_2,\Phi_3\right)$, 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 $U\subset\mathbb{C}$ or, more general, on a Riemann surface $\Sigma$. Most of them are natural in the sense of 'natural operators' as considered in \cite{KolarSlovakMichor}. The special case $d(z)=0$ related to minimal curves in $\mathbb{C}^3$ and minimal surfaces in $\mathbb{R}^3$ is of main interest. We start with the invariant construction of a sequence ${\bf V}^{(n)}$ of natural operators assigning to each pair $({\bf f},{\bf h})$ consisting of a meromorphic function $\bf f$ and a meromorphic vector field $\bf h$ on $\Sigma$ a minimal curve ${\bf V}^{(n)}({\bf f},{\bf h}):\Sigma\to\mathbb{C}^3$. The operator ${\bf V}^{(3)}$ is bijective and equivariant on a generic set of pairs $({\bf f},{\bf h})$. Algebraic representation formulas of minimal surfaces that arise from evolutes and caustics of curves in $\mathbb{R}^2$ 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.