SOME REMARKS ON TOTAL CURVATURE OF A MINIMAL GRAPH

Abstract

Abstract. In this paper we discuss bounds for the total curvature ofnonparametric minimal surfaces by using the properties of planar har-monic mappings. 1. IntroductionA complex–valued function f(z) = u(z) + iv(z), defined on some domainD ⊂ Cis a planar harmonic mapping if the components u and v are real–valuedharmonic functions which need not be conjugate. Throughout this article wewill discuss harmonic mappings of the unit disk D = {z ∈ C : |z| < 1}.A harmonic mapping f defined on D can be uniquely written as f = h + g,g(0) = 0, where h and g belong to the linear space H(D) of all holomorphicfunctions on D. The mapping f is locally univalent if and only if its Jacobian|h ′ | 2 − |g ′ | 2 does not vanish. If we require that f is orientation-preserving,then the second complex dilatation ω(z) = g ′ (z)/h ′ (z) belongs to H(D) and|ω(z)| < 1 on D. References for this material include [2] and [5].Harmonic univalent mappings were first studied in connection with minimalsurfaces by E. Heinz (see [7]). This relationship between a univalent harmonicmapping and a minimal graph M comes from conformal representation of Mvia the Weierstrass-Enneper formulas (see e.g. [4]). Let M = {(u,v,F(u,v)) :(u,v) ∈ Ω} be a nonparametric surface lying over a simply connected propersubdomain Ω of the complex plane C. If M is parametrized by orientation-preserving isothermal parameters z = x + iy ∈ D, the projection onto itsbase plane gives a univalent harmonic mapping f(z) = u + iv of D onto Ωwhose dilatation ω is the square of an holomorphic function with |ω(z)| < 1on D. Conversely, if f = h+g is an orientation-preserving univalent harmonicmapping of Donto Ω with dilatation ω = p