Complete minimal surfaces and harmonic functions

Abstract

We prove that for any open Riemann surface $M$ and any non constant harmonic function $h:M\to \mathbb{R},$ there exists a complete conformal minimal immersion $X:M\to {\mathbb{R}}^3$ 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.

10 pages

References in corpus (2)

• Existence of proper minimal surfaces of arbitrary topological type
• Complete minimal surfaces in R3 with a prescribed coordinate function