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.

我们得出了一个非常普遍的条件，即在单位圆盘\（{{{\ mathbb {D}}} \）中将平方扩张成可感测的保谐谐波映射，并允许对扩展复平面准拟形扩展。该分析取决于Weierstrass–Enneper升程到扩展平面的几何特性，该扩展平面将参数化的最小表面粘合到互补的拓扑半球上。只要满足对扩张的其他限制，所得的拓扑球体将在复杂平面上绘制整个图形。投影导致所需的扩展。从一般标准中得出了几个推论。