Abstract
We derive a very general condition for a sense-preserving harmonic mapping with dilatation a square to be injective in the unit disk $${{\mathbb {D}}}$$ and to admit a quasiconformal extension to the extended complex plane. The analysis depends on geometric properties of an extension of the Weierstrass–Enneper lift to the extended plane that glues the parametrized minimal surface to a complementary topological hemisphere. The resulting topological sphere renders an entire graph over the complex plane provided additional restrictions on the dilatation are satisfied. The projection results in the desired extension. Several corollaries are drawn from the general criterion.
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