Complete minimal surfaces and harmonic functions

Details Complete minimal surfaces and harmonic functions Complete minimal surfaces and harmonic functions

2009-10-22

arxiv.org/abs/0910.4284v1

Mathematics

Differential Geometry

10 pages

Scientific paper

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.

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