Bifurkation von Minimalflächen und elementare Katastrophen. (Bifurcation of minimal surfaces and elementary catastrophes) (Q1093219)

Let B be the unit open ball in \({\mathbb{R}}^ 2\), \(\Omega_ 0\) an open neighbourhood of \(\bar B,\) \(h: \Omega\) \({}_ 0\to {\mathbb{C}}^ a \)nonzero holomorphic function and \(f_ 0: B\to {\mathbb{R}}^ 3\) the representation of a minimal surface, where \[ f_ 0(x)^ T=Re\int^{x^ 1+ix^ 2}_{0} h(z)(1-z^ 2,i+iz^ 2,2z)^ T dz. \] The author deduces conditions on the coefficients of \(1/h(z)=\sum^{\infty}_{0}c_ nz^ n\) under which it is possible to describe the bifurcation of minimal surfaces \(f_ r\) from \(f_ 0\) (with \(\gamma_ r=f_ r|_{\partial B}\) being a homotopy of \(\gamma_ 0=f_ 0|_{\partial B})\) in terms of the fold, cusp and swallowtail catastrophe.