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- W2237384732 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." @default.
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- W2237384732 date "1988-01-01" @default.
- W2237384732 modified "2023-09-27" @default.
- W2237384732 title "Curvature Estimates for some Minimal Surfaces" @default.
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- W2237384732 doi "https://doi.org/10.1007/978-3-0348-9158-5_8" @default.
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