On harmonic quasiconformal immersions of surfaces in \(\mathbb R^3\) (Q2839380)

Motivated by the fact that a conformal parametrization of a surface in \(\mathbb{R}^3\) is harmonic if and only if the surface is minimal (in which case the Gauss map is also conformal), the authors study harmonic immersions of Riemann surfaces \(X:M\longrightarrow\mathbb{R}^3\) with quasiconformal Gauss map. They introduce a Weierstaß-type representation for harmonic immersions to study their geometric and analytic properties, and prove several results: They classify complete harmonic immersions of finite total curvature; show the existence of complete harmonic embeddings with geometry that is not shared by complete embedded minimal surfaces; extend a result of T. Weinstein [\textit{T. Klotz Milnor}, Proc. Am. Math. Soc. 78, 269--275 (1980; Zbl 0443.53003)] on the range of the Gauss map of harmonic immersions; prove, under suitable assumptions, that a harmonic immersion asymptotically conformal on the ideal boundary of M is conformal (i.e. minimal).