Commentarii Mathematici Helvetici

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Volume 87, Issue 4, 2012, pp. 891–904
DOI: 10.4171/CMH/272

Published online: 2012-10-10

Complete minimal surfaces and harmonic functions

Antonio Alarcón[1], Isabel Fernández[2] and Francisco J. López[3]

(1) Universidad de Granada, Spain
(2) Universidad de Sevilla, Spain
(3) Universidad de Granada, Spain

We prove that for any open Riemann surface $\mathcal{N}$ and any non-constant harmonic function $h\colon \mathcal{N} \to \mathbb{R}$, there exists a complete conformal minimal immersion $X\colon \mathcal{N} \to \mathbb{R}^3$ whose third coordinate function coincides with $h$.

As a consequence, complete minimal surfaces with arbitrary conformal structure and whose Gauss map misses two points are constructed.

Keywords: Complete minimal surfaces, harmonic functions on Riemann surfaces, Gauss map, holomorphic immersions

Alarcón Antonio, Fernández Isabel, López Francisco: Complete minimal surfaces and harmonic functions. Comment. Math. Helv. 87 (2012), 891-904. doi: 10.4171/CMH/272