Narrow Results

We obtain a characterization of complex curves in Kähler surfaces. In addition, by a direct method, we see that in a hyperKähler 4-manifold, complex curves are just isotropic minimal surfaces compatible with the orientation of the space. We obtain characterizations of hyperKähler 4-manifolds among Kähler surfaces, in terms of isotropic minimal surfaces and deformations of the complex structures.

In this paper, we obtain a local characterization of a complex curve in $C^2$ in terms of the induced metric and a holomorphic cubic differential $\mathrm{\Phi }$, and we see that any complex curve in $C^2$ is locally represented as the image by the composition of an affine Schwarz map and a parallel translation in $C^2$. We obtain a local characterization of a sphere Schwarz immersion in terms of a positive-valued function $\rho$.