On harmonic quasiconformal immersions of surfaces in $\mathbb{R}^3$

TL;DR

This paper investigates the global geometric properties of harmonic immersions of Riemann surfaces in three-dimensional space, emphasizing those with quasiconformal Gauss maps and finite total curvature, and introduces new examples with notable geometry.

Contribution

It provides new insights into the geometry of harmonic immersions with quasiconformal Gauss maps and constructs novel examples with significant geometric features.

Findings

Characterization of complete harmonic immersions with quasiconformal Gauss maps

Analysis of finite total curvature harmonic immersions

Construction of new examples with complex geometry

Abstract

This paper is devoted to the study of the global properties of harmonically immersed Riemann surfaces in $\mathbb{R}^3.$ We focus on the geometry of complete harmonic immersions with quasiconformal Gauss map, and in particular, of those with finite total curvature. We pay special attention to the construction of new examples with significant geometry.