Curvature Estimates for some Minimal Surfaces

Abstract

Let D be a simply-connected domain in the complex plane ℂ, and let f = u + iv be a univalent, orientation-preserving, harmonic mapping from D into ℂ. Then f can be written in the form f = h + $\bar g$ where h and g belong to the linear space H(D) of analytic functions on D. In addition, f can be viewed as a solution of the elliptic partial differential equation

$$ {\bar f_{\bar z}} = a{f_z} $$

(1.1)

where the function a = g′/h′ belongs to H(D) and satisfies |a(z)| < 1 for all z ∈ D. Hence the mapping f is locally quasiconformal. Conversely, any univalent solution of (1.1) with a analytic and |a| < 1 is an orientation-preserving harmonic mapping of D (see [4]). Observe that if φ is a conformal mapping from D ₁ onto D, then f ₁ = f o φ is a harmonic mapping defined on D ₁ with D ₁ with a ₁ = a o φ.