Complete minimal surfaces and harmonic functions

  • Antonio Alarcón

    Universidad de Granada, Spain
  • Isabel Fernández

    Universidad de Sevilla, Spain
  • Francisco J. López

    Universidad de Granada, Spain


We prove that for any open Riemann surface N\mathcal{N} and any non-constant harmonic function h ⁣:NRh\colon \mathcal{N} \to \mathbb{R}, there exists a complete conformal minimal immersion X ⁣:NR3X\colon \mathcal{N} \to \mathbb{R}^3 whose third coordinate function coincides with hh.

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. 891–904

DOI 10.4171/CMH/272