We prove that for any open Riemann surface N and any non constant harmonic function
h : N → R, there exists a complete conformal minimal immersion X : N → R3 whose third
coordinate function coincides with h.
As a consequence, completeminimal surfaceswith arbitrary conformal structure andwhose Gauss
map misses two points are constructed.Ministerio de Educación y Ciencia MTM2007-61775Ministerio de Educación y Ciencia MTM2007-64504Junta de Andalucía P06-FQM-01642
