Abstract
A maximal surface in the 3-dimensional Minkowski space L3 has the Weierstrass representation with a holomorphic function and a meromorphic function. We consider a maximal surface as a graph on the range of a harmonic mapping defined on the unit disk in the complex plane. We first verify a relation between the regularity of the maximal graph in L3 and the k-quasiconformality of the harmonic mapping. Second, we provide estimates of the Gaussian curvature and total curvature of a maximal graph in L3 using the canonical decomposition of a harmonic mapping. Third, we construct maximal graphs in L3 on several domains such as a horizontal-strip domain, a half-plane domain, a half-strip domain, a slit domain and regular n-polygons by using the shear construction for harmonic mappings.
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