Kennzeichnungen von Minimalschraubfl�chen Mittels Vertikalisophoten und Horizontalschattengrenzen

Abstract

In Euclidean space E3, let Φ be a (regular Cω-) minimal surface without planar points having locally (without loss of generality) the spherical representation n(u,v)=(cos v/cosh u, sin v/cosh u, tanh u), (u,v)eG⊂ℝ2. The corresponding (isothermal) parametrization Φ: x(u,v), (u,v)eG can be expressed using agenerating Function ψ(u,v) which satisfies ψuu +ψ vv − 2ψutanh u + ψ=0; the v-curves (coordinate curves u=u0) in Φ, along each of which the angle between the normal n(u,v) of Φ and the x3-axis is constant, are thevertical- isophotes of Φ, the u-curves (v=v0) being their orthogonal trajectories (theorems 1, 2). Considering u-curves and/or v-curves of Φ having additional geometric properties (curves of constant/steepest slope, curves of constant Gaussian curvature, asymptotic curves, lines of curvature or geodesies of Φ) we prove many newgeometric characterizations of theright helicoid, thecatenoid andScherk's second surface (theorems 3–7). All of these surfaces areminimal helicoidal surfaces.