We prove that for any open Riemann surface and any non-constant harmonic function , there exists a complete conformal minimal immersion whose third coordinate function coincides with .
As a consequence, complete minimal surfaces with arbitrary conformal structure and whose Gauss map misses two points are constructed.
Cite this article
Antonio Alarcón, Isabel Fernández, Francisco J. López, Complete minimal surfaces and harmonic functions. Comment. Math. Helv. 87 (2012), no. 4, pp. 891–904DOI 10.4171/CMH/272