Abstract
In this paper, we obtain a local characterization of a complex curve in [Formula: see text] in terms of the induced metric and a holomorphic cubic differential [Formula: see text], and we see that any complex curve in [Formula: see text] is locally represented as the image by the composition of an affine Schwarz map and a parallel translation in [Formula: see text]. We obtain a local characterization of a sphere Schwarz immersion in terms of a positive-valued function [Formula: see text].
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